id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-011001 | Euclidean Geometry: Heron's Formula | 6 | Question: A triangle has side lengths $a=8$, $b=45$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the ... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{8+45+49}{2}=51$.",
"Step 2: Compute $s-a=43$, $s-b=6$, $s-c=2$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=26316$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{26316}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{26316}$.) |
math-011002 | Geometry: Area Invariants | 6 | Work carefully and justify each inference: A triangle has side lengths $a=40$, $b=55$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you u... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{40+55+59}{2}=77$.",
"Step 2: Compute $s-a=37$, $s-b=22$, $s-c=18$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=1128204$... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{1128204}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations mus... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{1128204}$.) |
math-011003 | Geometry: Area Invariants | 6 | Show all reasoning: A triangle has side lengths $a=14$, $b=49$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{14+49+58}{2}=\\frac{121}{2}$.",
"Step 2: Compute $s-a=\\frac{93}{2}$, $s-b=\\frac{23}{2}$, $s-c=\\frac{5}{2}$.",
"... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{1294095}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{1294095}$.) |
math-011004 | Geometry: Side-Length Data to Area | 6 | Answer with a short justification: A triangle has side lengths $a=16$, $b=44$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{16+44+49}{2}=\\frac{109}{2}$.",
"Step 2: Compute $s-a=\\frac{77}{2}$, $s-b=\\frac{21}{2}$, $s-c=\\frac{11}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{1938783}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{1938783}$.) |
math-011005 | Geometry: Area Invariants | 6 | Answer with a short justification: A triangle has side lengths $a=12$, $b=36$, $c=46$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{12+36+46}{2}=47$.",
"Step 2: Compute $s-a=35$, $s-b=11$, $s-c=1$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=18095$.",... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{18095}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{18095}$.) |
math-011006 | Geometry: Area Invariants | 6 | Do not skip justification steps: A triangle has side lengths $a=35$, $b=47$, $c=51$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordin... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(51,0)$ so $AB=c=51$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{10130211}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computat... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{10130211}$.) |
math-011007 | Euclidean Geometry: Heron's Formula | 6 | Show all reasoning: A triangle has side lengths $a=23$, $b=41$, $c=45$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{23+41+45}{2}=\\frac{109}{2}$.",
"Step 2: Compute $s-a=\\frac{63}{2}$, $s-b=\\frac{27}{2}$, $s-c=\\frac{19}{2}$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{3522771}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{3522771}$.) |
math-011008 | Geometry: Side-Length Data to Area | 6 | Answer with a short justification: A triangle has side lengths $a=10$, $b=45$, $c=50$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{10+45+50}{2}=\\frac{105}{2}$.",
"Step 2: Compute $s-a=\\frac{85}{2}$, $s-b=\\frac{15}{2}$, $s-c=\\frac{5}{2}$.",
"... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{669375}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computatio... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{669375}$.) |
math-011009 | Geometry: Side-Length Data to Area | 6 | Track quantifiers carefully: A triangle has side lengths $a=35$, $b=40$, $c=51$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(51,0)$ so $AB=c=51$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{486864}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011010 | Geometry: Side-Length Data to Area | 6 | Determine the requested value: A triangle has side lengths $a=49$, $b=60$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(60,0)$ so $AB=c=60$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{28809599}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{28809599}$.) |
math-011011 | Geometry: Side-Length Data to Area | 6 | Track quantifiers carefully: A triangle has side lengths $a=47$, $b=48$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(48,0)$ so $AB=c=48$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{15478463}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both co... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{15478463}$.) |
math-011012 | Coordinate Geometry: Base–Height via Distances | 6 | Where appropriate, name the theorem you use: A triangle has side lengths $a=38$, $b=39$, $c=52$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{38+39+52}{2}=\\frac{129}{2}$.",
"Step 2: Compute $s-a=\\frac{53}{2}$, $s-b=\\frac{51}{2}$, $s-c=\\frac{25}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{8717175}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{8717175}$.) |
math-011013 | Coordinate Geometry: Base–Height via Distances | 6 | Make each step logically reversible (or explain if not): A triangle has side lengths $a=21$, $b=24$, $c=42$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side leng... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{21+24+42}{2}=\\frac{87}{2}$.",
"Step 2: Compute $s-a=\\frac{45}{2}$, $s-b=\\frac{39}{2}$, $s-c=\\frac{3}{2}$.",
"S... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{458055}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computatio... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{458055}$.) |
math-011014 | Euclidean Geometry: Heron's Formula | 6 | Give an answer and a quick verification: A triangle has side lengths $a=36$, $b=38$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(59,0)$ so $AB=c=59$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{6936615}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{6936615}$.) |
math-011015 | Geometry: Area Invariants | 6 | Determine the requested value: A triangle has side lengths $a=36$, $b=54$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{36+54+58}{2}=74$.",
"Step 2: Compute $s-a=38$, $s-b=20$, $s-c=16$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=899840$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{899840}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011016 | Geometry: Area Invariants | 6 | Proceed methodically: A triangle has side lengths $a=21$, $b=56$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(58,0)$ so $AB=c=58$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{5486535}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{5486535}$.) |
math-011017 | Geometry: Side-Length Data to Area | 6 | Warm-up: A triangle has side lengths $a=9$, $b=14$, $c=17$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the p... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(17,0)$ so $AB=c=17$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{3960}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must m... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{3960}$.) |
math-011018 | Euclidean Geometry: Heron's Formula | 6 | Answer using clear logical steps: A triangle has side lengths $a=31$, $b=54$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordi... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{31+54+56}{2}=\\frac{141}{2}$.",
"Step 2: Compute $s-a=\\frac{79}{2}$, $s-b=\\frac{33}{2}$, $s-c=\\frac{29}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{10660023}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both co... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011019 | Geometry: Area Invariants | 6 | Complete the analysis: A triangle has side lengths $a=18$, $b=34$, $c=34$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clea... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{18+34+34}{2}=43$.",
"Step 2: Compute $s-a=25$, $s-b=9$, $s-c=9$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=87075$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{87075}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{87075}$.) |
math-011020 | Coordinate Geometry: Base–Height via Distances | 6 | Carefully track domains: A triangle has side lengths $a=21$, $b=36$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cl... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{21+36+49}{2}=53$.",
"Step 2: Compute $s-a=32$, $s-b=17$, $s-c=4$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=115328$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{115328}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011021 | Geometry: Side-Length Data to Area | 6 | Write the solution set clearly: A triangle has side lengths $a=26$, $b=43$, $c=54$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(54,0)$ so $AB=c=54$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{4846815}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011022 | Geometry: Side-Length Data to Area | 6 | Provide both a computational and a conceptual explanation: A triangle has side lengths $a=26$, $b=34$, $c=53$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side le... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{26+34+53}{2}=\\frac{113}{2}$.",
"Step 2: Compute $s-a=\\frac{61}{2}$, $s-b=\\frac{45}{2}$, $s-c=\\frac{7}{2}$.",
"... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{2171295}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{2171295}$.) |
math-011023 | Euclidean Geometry: Heron's Formula | 6 | Solve and include a self-check: A triangle has side lengths $a=19$, $b=25$, $c=31$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(31,0)$ so $AB=c=31$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{901875}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both comp... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{901875}$.) |
math-011024 | Geometry: Side-Length Data to Area | 6 | Use two approaches if possible: A triangle has side lengths $a=30$, $b=40$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(48,0)$ so $AB=c=48$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{357599}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011025 | Coordinate Geometry: Base–Height via Distances | 6 | Exercise: A triangle has side lengths $a=39$, $b=41$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(55,0)$ so $AB=c=55$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{10195875}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant un... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{10195875}$.) |
math-011026 | Geometry: Side-Length Data to Area | 6 | Find the exact value: A triangle has side lengths $a=18$, $b=44$, $c=46$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{18+44+46}{2}=54$.",
"Step 2: Compute $s-a=36$, $s-b=10$, $s-c=8$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=155520$."... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{155520}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions,... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{155520}$.) |
math-011027 | Geometry: Area Invariants | 6 | Challenge: A triangle has side lengths $a=15$, $b=41$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state th... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{15+41+49}{2}=\\frac{105}{2}$.",
"Step 2: Compute $s-a=\\frac{75}{2}$, $s-b=\\frac{23}{2}$, $s-c=\\frac{7}{2}$.",
"... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{1267875}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011028 | Geometry: Side-Length Data to Area | 6 | Give a fully justified solution: A triangle has side lengths $a=44$, $b=44$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordin... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{44+44+48}{2}=68$.",
"Step 2: Compute $s-a=24$, $s-b=24$, $s-c=20$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=783360$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{783360}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{783360}$.) |
math-011029 | Geometry: Area Invariants | 6 | Checkpoint: A triangle has side lengths $a=36$, $b=49$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state t... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(59,0)$ so $AB=c=59$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{775008}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011030 | Geometry: Area Invariants | 6 | Track units/moduli carefully: A triangle has side lengths $a=31$, $b=52$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinate... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{31+52+60}{2}=\\frac{143}{2}$.",
"Step 2: Compute $s-a=\\frac{81}{2}$, $s-b=\\frac{39}{2}$, $s-c=\\frac{23}{2}$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{10389951}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computat... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{10389951}$.) |
math-011031 | Coordinate Geometry: Base–Height via Distances | 6 | Indicate where a theorem is used: A triangle has side lengths $a=36$, $b=45$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordi... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{36+45+48}{2}=\\frac{129}{2}$.",
"Step 2: Compute $s-a=\\frac{57}{2}$, $s-b=\\frac{39}{2}$, $s-c=\\frac{33}{2}$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{9463311}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{9463311}$.) |
math-011032 | Coordinate Geometry: Base–Height via Distances | 6 | Be explicit about assumptions: A triangle has side lengths $a=13$, $b=40$, $c=41$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(41,0)$ so $AB=c=41$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{67116}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011033 | Coordinate Geometry: Base–Height via Distances | 6 | Solve and justify each step: A triangle has side lengths $a=15$, $b=33$, $c=37$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(37,0)$ so $AB=c=37$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{977075}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computatio... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{977075}$.) |
math-011034 | Geometry: Area Invariants | 6 | Where appropriate, name the theorem you use: A triangle has side lengths $a=37$, $b=39$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{37+39+55}{2}=\\frac{131}{2}$.",
"Step 2: Compute $s-a=\\frac{57}{2}$, $s-b=\\frac{53}{2}$, $s-c=\\frac{21}{2}$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{8310771}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011035 | Geometry: Area Invariants | 6 | Try to avoid pattern-matching; explain why: A triangle has side lengths $a=25$, $b=42$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you ... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(59,0)$ so $AB=c=59$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{201096}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{201096}$.) |
math-011036 | Geometry: Side-Length Data to Area | 6 | State any required conditions first: A triangle has side lengths $a=29$, $b=38$, $c=47$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coo... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{29+38+47}{2}=57$.",
"Step 2: Compute $s-a=28$, $s-b=19$, $s-c=10$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=303240$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{303240}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{303240}$.) |
math-011037 | Coordinate Geometry: Base–Height via Distances | 6 | Work carefully and justify each inference: A triangle has side lengths $a=10$, $b=35$, $c=41$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you u... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{10+35+41}{2}=43$.",
"Step 2: Compute $s-a=33$, $s-b=8$, $s-c=2$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=22704$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{22704}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011038 | Geometry: Area Invariants | 6 | Give reasoning, not just computation: A triangle has side lengths $a=26$, $b=44$, $c=45$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use co... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(45,0)$ so $AB=c=45$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{4890375}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011039 | Geometry: Area Invariants | 6 | Compute the requested quantity: A triangle has side lengths $a=54$, $b=57$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{54+57+60}{2}=\\frac{171}{2}$.",
"Step 2: Compute $s-a=\\frac{63}{2}$, $s-b=\\frac{57}{2}$, $s-c=\\frac{51}{2}$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{31317111}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{31317111}$.) |
math-011040 | Geometry: Side-Length Data to Area | 6 | Carefully track domains: A triangle has side lengths $a=32$, $b=34$, $c=35$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cl... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{32+34+35}{2}=\\frac{101}{2}$.",
"Step 2: Compute $s-a=\\frac{37}{2}$, $s-b=\\frac{33}{2}$, $s-c=\\frac{31}{2}$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{3822951}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{3822951}$.) |
math-011041 | Geometry: Area Invariants | 6 | Find the exact value: A triangle has side lengths $a=24$, $b=35$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(58,0)$ so $AB=c=58$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{379431}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant unde... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{379431}$.) |
math-011042 | Euclidean Geometry: Heron's Formula | 6 | Solve (and briefly cross-validate): A triangle has side lengths $a=10$, $b=50$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coor... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{10+50+58}{2}=59$.",
"Step 2: Compute $s-a=49$, $s-b=9$, $s-c=1$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=26019$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{26019}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{26019}$.) |
math-011043 | Geometry: Side-Length Data to Area | 6 | Carefully track domains: A triangle has side lengths $a=9$, $b=43$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cle... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{9+43+48}{2}=50$.",
"Step 2: Compute $s-a=41$, $s-b=7$, $s-c=2$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=28700$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{28700}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{28700}$.) |
math-011044 | Geometry: Side-Length Data to Area | 6 | Provide a rigorous solution: A triangle has side lengths $a=52$, $b=54$, $c=57$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{52+54+57}{2}=\\frac{163}{2}$.",
"Step 2: Compute $s-a=\\frac{59}{2}$, $s-b=\\frac{55}{2}$, $s-c=\\frac{49}{2}$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{25917815}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant un... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{25917815}$.) |
math-011045 | Euclidean Geometry: Heron's Formula | 6 | Proceed methodically: A triangle has side lengths $a=23$, $b=41$, $c=43$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{23+41+43}{2}=\\frac{107}{2}$.",
"Step 2: Compute $s-a=\\frac{61}{2}$, $s-b=\\frac{25}{2}$, $s-c=\\frac{21}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{3426675}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{3426675}$.) |
math-011046 | Geometry: Area Invariants | 6 | Find the exact value: A triangle has side lengths $a=23$, $b=50$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(56,0)$ so $AB=c=56$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{5278551}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011047 | Geometry: Area Invariants | 6 | Try to avoid pattern-matching; explain why: A triangle has side lengths $a=17$, $b=51$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you ... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(55,0)$ so $AB=c=55$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{2988531}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{2988531}$.) |
math-011048 | Geometry: Side-Length Data to Area | 6 | Explain why your operations are valid: A triangle has side lengths $a=21$, $b=32$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use c... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(49,0)$ so $AB=c=49$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{58140}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{58140}$.) |
math-011049 | Euclidean Geometry: Heron's Formula | 6 | Use two approaches if possible: A triangle has side lengths $a=43$, $b=49$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(58,0)$ so $AB=c=58$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{1060800}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigi... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{1060800}$.) |
math-011050 | Geometry: Side-Length Data to Area | 6 | Solve (and briefly cross-validate): A triangle has side lengths $a=26$, $b=27$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coor... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(49,0)$ so $AB=c=49$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{61200}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{61200}$.) |
math-011051 | Geometry: Side-Length Data to Area | 6 | Solve with verification: A triangle has side lengths $a=17$, $b=36$, $c=43$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cl... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(43,0)$ so $AB=c=43$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{89280}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011052 | Geometry: Side-Length Data to Area | 6 | Exercise: A triangle has side lengths $a=6$, $b=27$, $c=30$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the ... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(30,0)$ so $AB=c=30$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{86751}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011053 | Geometry: Area Invariants | 6 | Answer with a short justification: A triangle has side lengths $a=23$, $b=36$, $c=43$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(43,0)$ so $AB=c=43$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{171360}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{171360}$.) |
math-011054 | Euclidean Geometry: Heron's Formula | 6 | Task: A triangle has side lengths $a=40$, $b=43$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the poi... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(55,0)$ so $AB=c=55$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{728364}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions,... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{728364}$.) |
math-011055 | Geometry: Side-Length Data to Area | 6 | Checkpoint: A triangle has side lengths $a=25$, $b=48$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state t... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(49,0)$ so $AB=c=49$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{342576}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011056 | Euclidean Geometry: Heron's Formula | 6 | Track units/moduli carefully: A triangle has side lengths $a=12$, $b=45$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinate... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{12+45+49}{2}=53$.",
"Step 2: Compute $s-a=41$, $s-b=8$, $s-c=4$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=69536$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{69536}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{69536}$.) |
math-011057 | Coordinate Geometry: Base–Height via Distances | 6 | Compute the requested quantity: A triangle has side lengths $a=13$, $b=35$, $c=36$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{13+35+36}{2}=42$.",
"Step 2: Compute $s-a=29$, $s-b=7$, $s-c=6$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=51156$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{51156}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{51156}$.) |
math-011058 | Coordinate Geometry: Base–Height via Distances | 6 | Keep the final answer in boxed form: A triangle has side lengths $a=44$, $b=54$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coo... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(58,0)$ so $AB=c=58$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{1272960}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computatio... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{1272960}$.) |
math-011059 | Geometry: Side-Length Data to Area | 6 | Give a fully justified solution: A triangle has side lengths $a=21$, $b=47$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordin... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{21+47+55}{2}=\\frac{123}{2}$.",
"Step 2: Compute $s-a=\\frac{81}{2}$, $s-b=\\frac{29}{2}$, $s-c=\\frac{13}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{3756051}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{3756051}$.) |
math-011060 | Geometry: Side-Length Data to Area | 6 | Solve with verification: A triangle has side lengths $a=32$, $b=38$, $c=57$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cl... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{32+38+57}{2}=\\frac{127}{2}$.",
"Step 2: Compute $s-a=\\frac{63}{2}$, $s-b=\\frac{51}{2}$, $s-c=\\frac{13}{2}$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{5304663}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011061 | Euclidean Geometry: Heron's Formula | 6 | Proceed methodically: A triangle has side lengths $a=16$, $b=42$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{16+42+56}{2}=57$.",
"Step 2: Compute $s-a=41$, $s-b=15$, $s-c=1$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=35055$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{35055}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{35055}$.) |
math-011062 | Coordinate Geometry: Base–Height via Distances | 6 | Proceed methodically: A triangle has side lengths $a=39$, $b=52$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(59,0)$ so $AB=c=59$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{993600}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011063 | Coordinate Geometry: Base–Height via Distances | 6 | Give a theorem-based solution: A triangle has side lengths $a=35$, $b=42$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(60,0)$ so $AB=c=60$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{8270279}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011064 | Geometry: Side-Length Data to Area | 6 | Derive the result step-by-step: A triangle has side lengths $a=20$, $b=46$, $c=52$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(52,0)$ so $AB=c=52$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{209391}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{209391}$.) |
math-011065 | Coordinate Geometry: Base–Height via Distances | 6 | Exercise: A triangle has side lengths $a=41$, $b=43$, $c=52$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{41+43+52}{2}=68$.",
"Step 2: Compute $s-a=27$, $s-b=25$, $s-c=16$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=734400$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{734400}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{734400}$.) |
math-011066 | Coordinate Geometry: Base–Height via Distances | 6 | Answer with a short justification: A triangle has side lengths $a=41$, $b=55$, $c=57$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{41+55+57}{2}=\\frac{153}{2}$.",
"Step 2: Compute $s-a=\\frac{71}{2}$, $s-b=\\frac{43}{2}$, $s-c=\\frac{39}{2}$.",
... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{18217251}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant un... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011067 | Euclidean Geometry: Heron's Formula | 6 | Challenge: A triangle has side lengths $a=34$, $b=44$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state th... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{34+44+48}{2}=63$.",
"Step 2: Compute $s-a=29$, $s-b=19$, $s-c=15$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=520695$.... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{520695}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011068 | Coordinate Geometry: Base–Height via Distances | 6 | Make each step logically reversible (or explain if not): A triangle has side lengths $a=22$, $b=28$, $c=36$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side leng... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(36,0)$ so $AB=c=36$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{94815}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011069 | Geometry: Area Invariants | 6 | Track units/moduli carefully: A triangle has side lengths $a=27$, $b=42$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinate... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(48,0)$ so $AB=c=48$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{5108103}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011070 | Euclidean Geometry: Heron's Formula | 6 | Carefully track domains: A triangle has side lengths $a=29$, $b=47$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cl... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{29+47+58}{2}=67$.",
"Step 2: Compute $s-a=38$, $s-b=20$, $s-c=9$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=458280$."... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{458280}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computation... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011071 | Coordinate Geometry: Base–Height via Distances | 6 | Keep the final answer in boxed form: A triangle has side lengths $a=21$, $b=28$, $c=41$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coo... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(41,0)$ so $AB=c=41$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{73440}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011072 | Coordinate Geometry: Base–Height via Distances | 6 | Try to avoid pattern-matching; explain why: A triangle has side lengths $a=46$, $b=56$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you ... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{46+56+58}{2}=80$.",
"Step 2: Compute $s-a=34$, $s-b=24$, $s-c=22$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=1436160$... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{1436160}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigi... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{1436160}$.) |
math-011073 | Geometry: Side-Length Data to Area | 6 | Explain why your operations are valid: A triangle has side lengths $a=22$, $b=41$, $c=58$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use c... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(58,0)$ so $AB=c=58$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{1816815}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{1816815}$.) |
math-011074 | Euclidean Geometry: Heron's Formula | 6 | Task: A triangle has side lengths $a=17$, $b=33$, $c=35$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the poi... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(35,0)$ so $AB=c=35$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{1235475}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011075 | Geometry: Side-Length Data to Area | 6 | Where appropriate, name the theorem you use: A triangle has side lengths $a=27$, $b=32$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{27+32+56}{2}=\\frac{115}{2}$.",
"Step 2: Compute $s-a=\\frac{61}{2}$, $s-b=\\frac{51}{2}$, $s-c=\\frac{3}{2}$.",
"... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{1073295}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{1073295}$.) |
math-011076 | Geometry: Side-Length Data to Area | 6 | Warm-up: A triangle has side lengths $a=24$, $b=32$, $c=39$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the ... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(39,0)$ so $AB=c=39$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{\\frac{2353055}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computati... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{2353055}$.) |
math-011077 | Coordinate Geometry: Base–Height via Distances | 6 | Exercise: A triangle has side lengths $a=24$, $b=27$, $c=45$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(45,0)$ so $AB=c=45$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{72576}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011078 | Euclidean Geometry: Heron's Formula | 6 | Indicate where a theorem is used: A triangle has side lengths $a=17$, $b=20$, $c=23$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordi... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(23,0)$ so $AB=c=23$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{27300}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011079 | Geometry: Side-Length Data to Area | 6 | Be explicit about assumptions: A triangle has side lengths $a=10$, $b=32$, $c=33$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(33,0)$ so $AB=c=33$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{408375}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant unde... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{408375}$.) |
math-011080 | Coordinate Geometry: Base–Height via Distances | 6 | Where appropriate, name the theorem you use: A triangle has side lengths $a=25$, $b=46$, $c=46$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{25+46+46}{2}=\\frac{117}{2}$.",
"Step 2: Compute $s-a=\\frac{67}{2}$, $s-b=\\frac{25}{2}$, $s-c=\\frac{25}{2}$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{4899375}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{4899375}$.) |
math-011081 | Geometry: Side-Length Data to Area | 6 | Exercise: A triangle has side lengths $a=28$, $b=34$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(56,0)$ so $AB=c=56$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{137175}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions,... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011082 | Coordinate Geometry: Base–Height via Distances | 6 | Track quantifiers carefully: A triangle has side lengths $a=14$, $b=49$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(60,0)$ so $AB=c=60$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{876375}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid m... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{876375}$.) |
math-011083 | Geometry: Side-Length Data to Area | 6 | Challenge: A triangle has side lengths $a=47$, $b=59$, $c=60$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state th... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(60,0)$ so $AB=c=60$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{1649376}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{1649376}$.) |
math-011084 | Geometry: Side-Length Data to Area | 6 | Write the solution set clearly: A triangle has side lengths $a=28$, $b=29$, $c=48$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{28+29+48}{2}=\\frac{105}{2}$.",
"Step 2: Compute $s-a=\\frac{49}{2}$, $s-b=\\frac{47}{2}$, $s-c=\\frac{9}{2}$.",
"... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{2176335}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{2176335}$.) |
math-011085 | Geometry: Side-Length Data to Area | 6 | Answer with a short justification: A triangle has side lengths $a=42$, $b=47$, $c=51$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coord... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{42+47+51}{2}=70$.",
"Step 2: Compute $s-a=28$, $s-b=23$, $s-c=19$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=856520$.... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{856520}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Core principle: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011086 | Geometry: Area Invariants | 6 | Question: A triangle has side lengths $a=10$, $b=12$, $c=20$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{10+12+20}{2}=21$.",
"Step 2: Compute $s-a=11$, $s-b=9$, $s-c=1$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=2079$.",
... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{2079}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must m... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Takeaway: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011087 | Euclidean Geometry: Heron's Formula | 6 | Provide a rigorous solution: A triangle has side lengths $a=14$, $b=18$, $c=18$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(18,0)$ so $AB=c=18$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{13475}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011088 | Geometry: Side-Length Data to Area | 6 | Derive the result step-by-step: A triangle has side lengths $a=47$, $b=55$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{47+55+59}{2}=\\frac{161}{2}$.",
"Step 2: Compute $s-a=\\frac{67}{2}$, $s-b=\\frac{51}{2}$, $s-c=\\frac{43}{2}$.",
... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{23655891}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011089 | Coordinate Geometry: Base–Height via Distances | 6 | Use two approaches if possible: A triangle has side lengths $a=19$, $b=29$, $c=30$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{19+29+30}{2}=39$.",
"Step 2: Compute $s-a=20$, $s-b=10$, $s-c=9$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=70200$.",... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{70200}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{70200}$.) |
math-011090 | Geometry: Area Invariants | 6 | Solve (and briefly cross-validate): A triangle has side lengths $a=34$, $b=38$, $c=49$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coor... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{34+38+49}{2}=\\frac{121}{2}$.",
"Step 2: Compute $s-a=\\frac{53}{2}$, $s-b=\\frac{45}{2}$, $s-c=\\frac{23}{2}$.",
... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{6637455}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{6637455}$.) |
math-011091 | Geometry: Area Invariants | 6 | Proceed methodically: A triangle has side lengths $a=27$, $b=49$, $c=54$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(54,0)$ so $AB=c=54$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{434720}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{434720}$.) |
math-011092 | Geometry: Area Invariants | 6 | Question: A triangle has side lengths $a=11$, $b=59$, $c=59$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state the... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(59,0)$ so $AB=c=59$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{\\frac{1670163}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both com... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011093 | Coordinate Geometry: Base–Height via Distances | 6 | Find the exact value: A triangle has side lengths $a=21$, $b=30$, $c=31$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{21+30+31}{2}=41$.",
"Step 2: Compute $s-a=20$, $s-b=11$, $s-c=10$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=90200$."... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{90200}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011094 | Coordinate Geometry: Base–Height via Distances | 6 | Solve and sanity-check: A triangle has side lengths $a=33$, $b=45$, $c=55$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, cle... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(55,0)$ so $AB=c=55$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{\\frac{8812979}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant und... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011095 | Geometry: Side-Length Data to Area | 6 | Challenge: A triangle has side lengths $a=33$, $b=34$, $c=45$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clearly state th... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(45,0)$ so $AB=c=45$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{311696}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions,... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011096 | Geometry: Area Invariants | 6 | Solve and include a self-check: A triangle has side lengths $a=32$, $b=44$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordina... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(56,0)$ so $AB=c=56$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{493680}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{493680}$.) |
math-011097 | Euclidean Geometry: Heron's Formula | 6 | Indicate where a theorem is used: A triangle has side lengths $a=35$, $b=42$, $c=47$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordi... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{35+42+47}{2}=62$.",
"Step 2: Compute $s-a=27$, $s-b=20$, $s-c=15$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=502200$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{502200}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{502200}$.) |
math-011098 | Coordinate Geometry: Base–Height via Distances | 6 | Use two approaches if possible: A triangle has side lengths $a=9$, $b=36$, $c=37$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinat... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(37,0)$ so $AB=c=37$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{26240}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid ... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. |
math-011099 | Coordinate Geometry: Base–Height via Distances | 6 | Indicate where a theorem is used: A triangle has side lengths $a=11$, $b=38$, $c=44$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordi... | [
{
"method_name": "Coordinate Geometry (Distance Constraints)",
"approach": "Place one side on the $x$-axis, solve for the third vertex using distances, and compute $\\frac12\\cdot\\text{base}\\cdot\\text{height}$.",
"steps": [
"Step 1: Place $A=(0,0)$ and $B=(44,0)$ so $AB=c=44$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{\\frac{561255}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid m... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Key idea: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{\frac{561255}$.) |
math-011100 | Euclidean Geometry: Heron's Formula | 6 | Find the exact value: A triangle has side lengths $a=38$, $b=42$, $c=56$.
(a) Compute its area using Heron's formula.
(b) Give an independent derivation by placing the triangle in coordinates (base–height).
(c) Explain why the two methods must agree even though one uses only side lengths.
If you use coordinates, clear... | [
{
"method_name": "Heron's Formula",
"approach": "Compute semiperimeter $s$ and apply $[\\triangle]=\\sqrt{s(s-a)(s-b)(s-c)}$.",
"steps": [
"Step 1: Semiperimeter $s=\\frac{38+42+56}{2}=68$.",
"Step 2: Compute $s-a=30$, $s-b=26$, $s-c=12$.",
"Step 3: Multiply: $s(s-a)(s-b)(s-c)=636480$.... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{636480}$.\nHeron's formula is a side-length invariant. The coordinate method computes the same geometric area after embedding the triangle in the plane; since area is invariant under rigid motions, both computations must... | [
{
"error_description": "Used perimeter instead of semiperimeter in Heron's formula.",
"why_plausible": "The symbol $s$ is easy to confuse with the full perimeter.",
"why_wrong": "Heron's formula requires $s=(a+b+c)/2$; using $a+b+c$ squares the scale incorrectly.",
"which_method_catches_it": "Coordi... | Remember: The same geometric quantity can be computed from different invariants: side lengths (Heron) or coordinates (base–height). Agreement across methods is a powerful correctness check. (Here the result is $\boxed{\sqrt{636480}$.) |
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