id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-001001 | Coordinate Geometry: Distance Formula | 1 | Problem: Let $A(-152,79)$ and $B(-79,199)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicit... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(73,120)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{19729}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(73,120)$ as 19729, hence both give $AB=\\sqrt{19729}$.",
"robustness_analysis": "Robustness note: The v... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001002 | Coordinate Geometry: Distance Formula | 1 | Keep the final answer in boxed form: Let $A(-134,91)$ and $B(-166,-107)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or th... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-32,-198)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{40228}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-32,-198)$ as 40228, hence both give $AB=\\sqrt{40228}$.",
"robustness_analysis": "Robustness note: The vector/dot-product met... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{40228}$.) |
math-001003 | Geometry: Pythagorean Theorem in Coordinates | 1 | Answer using clear logical steps: Let $A(-106,-174)$ and $B(-28,-4)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Py... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(78,170)$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{34984}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(78,170)$ as 34984, hence both give $AB=\\sqrt{34984}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-pr... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{34984}$.) |
math-001004 | Coordinate Geometry: Distance Formula | 1 | Write the solution set clearly: Let $A(154,21)$ and $B(55,-11)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythago... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-99,-32)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{10825}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-99,-32)$ as 10825, hence both give $AB=\\sqrt{10825}$.",
"robustness_analysis": "Robustness note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001005 | Analytic Geometry: Translation Invariance | 1 | Solve and include a self-check: Let $A(-18,-45)$ and $B(-127,62)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-109,107)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{23330}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-109,107)$ as 23330, hence both give $AB=\\sqrt{23330}$.",
"robustness_analysis": "Robustness note: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001006 | Coordinate Geometry: Distance Formula | 1 | Where appropriate, name the theorem you use: Let $A(-82,192)$ and $B(42,-25)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=42-(-82)=124$ and $\\Delta y=-25-(192)=-217$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{62465}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(124,-217)$ as 62465, hence both give $AB=\\sqrt{62465}$.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{62465}$.) |
math-001007 | Coordinate Geometry: Distance Formula | 1 | Indicate where a theorem is used: Let $A(-15,-183)$ and $B(189,78)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=189-(-15)=204$ and $\\Delta y=78-(-183)=261$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{109737}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(204,261)$ as 109737, hence both give $AB=\\sqrt{109737}$.",
"robustness_analysis": "Robustness note: The vector/dot-pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{109737}$.) |
math-001008 | Vector Geometry: Norms and Dot Products | 1 | Indicate where a theorem is used: Let $A(24,-183)$ and $B(-14,113)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-14-(24)=-38$ and $\\Delta y=113-(-183)=296$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{89060}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-38,296)$ as 89060, hence both give $AB=\\sqrt{89060}$.",
"robustness_analysis": "Generality note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{89060}$.) |
math-001009 | Analytic Geometry: Translation Invariance | 1 | Find the exact value: Let $A(-36,70)$ and $B(-79,-184)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean the... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-43,-254)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{66365}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-43,-254)$ as 66365, hence both give $AB=\\sqrt{66365}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{66365}$.) |
math-001010 | Coordinate Geometry: Distance Formula | 1 | Provide both a computational and a conceptual explanation: Let $A(139,151)$ and $B(37,27)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should refer... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=37-(139)=-102$ and $\\Delta y=27-(151)=-124$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{25780}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-102,-124)$ as 25780, hence both give $AB=\\sqrt{25780}$.",
"robustness_analysis": "Robustness note: The vector/dot-prod... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{25780}$.) |
math-001011 | Coordinate Geometry: Distance Formula | 1 | Solve and then verify: Let $A(-55,-85)$ and $B(-75,-174)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean t... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-75-(-55)=-20$ and $\\Delta y=-174-(-85)=-89$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{8321}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-20,-89)$ as 8321, hence both give $AB=\\sqrt{8321}$.",
"robustness_analysis": "Robustness note: The vector/dot-product m... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{8321}$.) |
math-001012 | Coordinate Geometry: Distance Formula | 1 | Problem: Let $A(169,-80)$ and $B(-55,-153)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-224,-73)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{55505}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-224,-73)$ as 55505, hence both give $AB=\\sqrt{55505}$.",
"robustness_analysis": "If the problem were perturbed: The ve... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{55505}$.) |
math-001013 | Coordinate Geometry: Distance Formula | 1 | Explain each transformation: Let $A(-37,-117)$ and $B(114,-173)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(151,-56)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{25937}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(151,-56)$ as 25937, hence both give $AB=\\sqrt{25937}$.",
"robustness_analysis": "Generality note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001014 | Vector Geometry: Norms and Dot Products | 1 | Solve (and briefly cross-validate): Let $A(78,-145)$ and $B(167,97)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Py... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(89,242)$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{66485}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(89,242)$ as 66485, hence both give $AB=\\sqrt{66485}$.",
"robustness_analysis": "Generality note: The vector/dot-product... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001015 | Vector Geometry: Norms and Dot Products | 1 | Answer using clear logical steps: Let $A(169,-109)$ and $B(-94,-83)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Py... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-263,26)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{69845}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-263,26)$ as 69845, hence both give $AB=\\sqrt{69845}$.",
"robustness_analysis": "Robustness note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001016 | Coordinate Geometry: Distance Formula | 1 | Make each step logically reversible (or explain if not): Let $A(10,89)$ and $B(164,-181)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should refere... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=164-(10)=154$ and $\\Delta y=-181-(89)=-270$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{96616}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(154,-270)$ as 96616, hence both give $AB=\\sqrt{96616}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{96616}$.) |
math-001017 | Vector Geometry: Norms and Dot Products | 1 | Problem: Let $A(-14,62)$ and $B(137,-52)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitl... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(151,-114)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{35797}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(151,-114)$ as 35797, hence both give $AB=\\sqrt{35797}$.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{35797}$.) |
math-001018 | Analytic Geometry: Translation Invariance | 1 | Exercise: Let $A(27,-163)$ and $B(-181,33)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-208,196)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{81680}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-208,196)$ as 81680, hence both give $AB=\\sqrt{81680}$.",
"robustness_analysis": "Generality note: The vector/dot-produ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{81680}$.) |
math-001019 | Coordinate Geometry: Distance Formula | 1 | Use two approaches if possible: Let $A(-166,-153)$ and $B(-144,-171)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(22,-18)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{808}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(22,-18)$ as 808, hence both give $AB=\\sqrt{808}$.",
"robustness_analysis": "If the problem were perturbe... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001020 | Analytic Geometry: Translation Invariance | 1 | Prompt: Let $A(-61,20)$ and $B(-91,-9)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitly.... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-30,-29)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{1741}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-30,-29)$ as 1741, hence both give $AB=\\sqrt{1741}$.",
"robustness_analysis": "If the problem ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{1741}$.) |
math-001021 | Geometry: Pythagorean Theorem in Coordinates | 1 | Checkpoint: Let $A(-123,-92)$ and $B(129,-66)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem expl... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=129-(-123)=252$ and $\\Delta y=-66-(-92)=26$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{64180}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(252,26)$ as 64180, hence both give $AB=\\sqrt{64180}$.",
"robustness_analysis": "Sensitivity a... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{64180}$.) |
math-001022 | Vector Geometry: Norms and Dot Products | 1 | Provide a rigorous solution: Let $A(79,58)$ and $B(-79,44)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-158,-14)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{25160}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-158,-14)$ as 25160, hence both give $AB=\\sqrt{25160}$.",
"robustness_analysis": "Generality ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001023 | Coordinate Geometry: Distance Formula | 1 | Use two approaches if possible: Let $A(-187,59)$ and $B(200,-171)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(387,-230)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{202669}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(387,-230)$ as 202669, hence both give $AB=\\sqrt{202669}$.",
"robustness_analysis": "Robustne... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{202669}$.) |
math-001024 | Analytic Geometry: Translation Invariance | 1 | Provide both a computational and a conceptual explanation: Let $A(-36,-20)$ and $B(-26,-35)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should ref... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(10,-15)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{325}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(10,-15)$ as 325, hence both give $AB=\\sqrt{325}$.",
"robustness_analysis": "Sensitivity analysis: The ve... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{325}$.) |
math-001025 | Geometry: Pythagorean Theorem in Coordinates | 1 | Problem: Let $A(-70,-4)$ and $B(-146,20)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitl... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-76,24)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{6352}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-76,24)$ as 6352, hence both give $AB=\\sqrt{6352}$.",
"robustness_analysis": "Sensitivity analysis: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001026 | Analytic Geometry: Translation Invariance | 1 | Solve and then verify: Let $A(75,-76)$ and $B(146,36)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theo... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(71,112)$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{17585}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(71,112)$ as 17585, hence both give $AB=\\sqrt{17585}$.",
"robustness_analysis": "Generality note: The vector/dot-product... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001027 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve (and briefly cross-validate): Let $A(37,98)$ and $B(85,86)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(48,-12)$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{2448}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(48,-12)$ as 2448, hence both give $AB=\\sqrt{2448}$.",
"robustness_analysis": "If the problem were perturbed: The vector/... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001028 | Coordinate Geometry: Distance Formula | 1 | Provide both a computational and a conceptual explanation: Let $A(-36,32)$ and $B(108,-155)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should ref... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=108-(-36)=144$ and $\\Delta y=-155-(32)=-187$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{55705}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(144,-187)$ as 55705, hence both give $AB=\\sqrt{55705}$.",
"robustness_analysis": "If the problem were perturbed: The vector/d... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{55705}$.) |
math-001029 | Vector Geometry: Norms and Dot Products | 1 | Give a theorem-based solution: Let $A(90,123)$ and $B(38,117)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagor... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-52,-6)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{2740}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-52,-6)$ as 2740, hence both give $AB=\\sqrt{2740}$.",
"robustness_analysis": "Robustness note: The vector/dot-product method g... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001030 | Vector Geometry: Norms and Dot Products | 1 | Checkpoint: Let $A(-198,-93)$ and $B(-96,-70)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem expl... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-96-(-198)=102$ and $\\Delta y=-70-(-93)=23$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{10933}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(102,23)$ as 10933, hence both give $AB=\\sqrt{10933}$.",
"robustness_analysis": "Generality note: The v... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001031 | Coordinate Geometry: Distance Formula | 1 | Provide a rigorous solution: Let $A(55,-111)$ and $B(142,-7)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagore... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=142-(55)=87$ and $\\Delta y=-7-(-111)=104$.",
"Step 2: A translation sends ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{18385}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(87,104)$ as 18385, hence both give $AB=\\sqrt{18385}$.",
"robustness_analysis": "Robustness note: The v... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001032 | Vector Geometry: Norms and Dot Products | 1 | Start by stating any domain restrictions: Let $A(-53,152)$ and $B(77,-25)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=77-(-53)=130$ and $\\Delta y=-25-(152)=-177$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{48229}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(130,-177)$ as 48229, hence both give $AB=\\sqrt{48229}$.",
"robustness_analysis": "Robustness ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001033 | Analytic Geometry: Translation Invariance | 1 | Solve and sanity-check: Let $A(95,-172)$ and $B(-68,-64)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean t... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-163,108)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{38233}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-163,108)$ as 38233, hence both give $AB=\\sqrt{38233}$.",
"robustness_analysis": "Robustness note: The vector/dot-product met... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001034 | Geometry: Pythagorean Theorem in Coordinates | 1 | Answer using clear logical steps: Let $A(76,61)$ and $B(80,106)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=80-(76)=4$ and $\\Delta y=106-(61)=45$.",
"Step 2: A translation sends $A$ ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{2041}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(4,45)$ as 2041, hence both give $AB=\\sqrt{2041}$.",
"robustness_analysis": "Generality note: T... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001035 | Geometry: Pythagorean Theorem in Coordinates | 1 | Explain each transformation: Let $A(-95,28)$ and $B(-83,82)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorea... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(12,54)$.",
"Step 2: C... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{3060}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(12,54)$ as 3060, hence both give $AB=\\sqrt{3060}$.",
"robustness_analysis": "If the problem we... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{3060}$.) |
math-001036 | Analytic Geometry: Translation Invariance | 1 | Give a fully justified solution: Let $A(73,-112)$ and $B(-180,-38)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-180-(73)=-253$ and $\\Delta y=-38-(-112)=74$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{69485}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-253,74)$ as 69485, hence both give $AB=\\sqrt{69485}$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{69485}$.) |
math-001037 | Analytic Geometry: Translation Invariance | 1 | Show all reasoning: Let $A(47,-71)$ and $B(53,151)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=53-(47)=6$ and $\\Delta y=151-(-71)=222$.",
"Step 2: A translation sends $A... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{49320}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(6,222)$ as 49320, hence both give $AB=\\sqrt{49320}$.",
"robustness_analysis": "If the problem were perturbed: The vector/dot-... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{49320}$.) |
math-001038 | Geometry: Pythagorean Theorem in Coordinates | 1 | Where appropriate, name the theorem you use: Let $A(3,-131)$ and $B(64,-137)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors ... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(61,-6)$.",
"Step 2: C... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{3757}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(61,-6)$ as 3757, hence both give $AB=\\sqrt{3757}$.",
"robustness_analysis": "Robustness note: ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001039 | Vector Geometry: Norms and Dot Products | 1 | Start by stating any domain restrictions: Let $A(-18,80)$ and $B(34,-151)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=34-(-18)=52$ and $\\Delta y=-151-(80)=-231$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{56065}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(52,-231)$ as 56065, hence both give $AB=\\sqrt{56065}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001040 | Coordinate Geometry: Distance Formula | 1 | Where appropriate, name the theorem you use: Let $A(-90,171)$ and $B(119,-150)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vector... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=119-(-90)=209$ and $\\Delta y=-150-(171)=-321$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{146722}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(209,-321)$ as 146722, hence both give $AB=\\sqrt{146722}$.",
"robustness_analysis": "Generality note: The vector/dot-product ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{146722}$.) |
math-001041 | Coordinate Geometry: Distance Formula | 1 | Explain what is being counted/optimized: Let $A(-147,-65)$ and $B(113,115)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(260,180)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{100000}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(260,180)$ as 100000, hence both give $AB=\\sqrt{100000}$.",
"robustness_analysis": "If the problem were perturbed: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{100000}$.) |
math-001042 | Vector Geometry: Norms and Dot Products | 1 | Give an answer and a quick verification: Let $A(-50,122)$ and $B(39,105)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or t... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(89,-17)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{8210}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(89,-17)$ as 8210, hence both give $AB=\\sqrt{8210}$.",
"robustness_analysis": "Generality note: The vector/dot-product method g... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001043 | Vector Geometry: Norms and Dot Products | 1 | Question: Let $A(-34,-50)$ and $B(134,111)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=134-(-34)=168$ and $\\Delta y=111-(-50)=161$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{54145}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(168,161)$ as 54145, hence both give $AB=\\sqrt{54145}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001044 | Coordinate Geometry: Distance Formula | 1 | Solve (and briefly cross-validate): Let $A(-167,103)$ and $B(60,11)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Py... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=60-(-167)=227$ and $\\Delta y=11-(103)=-92$.",
"Step 2: A translation sends... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{59993}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(227,-92)$ as 59993, hence both give $AB=\\sqrt{59993}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001045 | Vector Geometry: Norms and Dot Products | 1 | Provide both a computational and a conceptual explanation: Let $A(-184,198)$ and $B(-89,-89)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should re... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-89-(-184)=95$ and $\\Delta y=-89-(198)=-287$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{91394}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(95,-287)$ as 91394, hence both give $AB=\\sqrt{91394}$.",
"robustness_analysis": "Robustness n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{91394}$.) |
math-001046 | Coordinate Geometry: Distance Formula | 1 | Track units/moduli carefully: Let $A(33,109)$ and $B(-191,125)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythago... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-191-(33)=-224$ and $\\Delta y=125-(109)=16$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{50432}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-224,16)$ as 50432, hence both give $AB=\\sqrt{50432}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{50432}$.) |
math-001047 | Coordinate Geometry: Distance Formula | 1 | Solve and include a self-check: Let $A(-16,121)$ and $B(178,-74)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=178-(-16)=194$ and $\\Delta y=-74-(121)=-195$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{75661}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(194,-195)$ as 75661, hence both give $AB=\\sqrt{75661}$.",
"robustness_analysis": "Generality note: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001048 | Geometry: Pythagorean Theorem in Coordinates | 1 | Write the solution set clearly: Let $A(24,30)$ and $B(177,-76)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythago... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(153,-106)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{34645}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(153,-106)$ as 34645, hence both give $AB=\\sqrt{34645}$.",
"robustness_analysis": "If the problem were perturbed: The vector/d... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{34645}$.) |
math-001049 | Coordinate Geometry: Distance Formula | 1 | Give an answer and a quick verification: Let $A(161,152)$ and $B(62,97)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or th... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=62-(161)=-99$ and $\\Delta y=97-(152)=-55$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{12826}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-99,-55)$ as 12826, hence both give $AB=\\sqrt{12826}$.",
"robustness_analysis": "If the problem were perturbed: The vector/do... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{12826}$.) |
math-001050 | Geometry: Pythagorean Theorem in Coordinates | 1 | Prompt: Let $A(156,-174)$ and $B(-10,39)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitl... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-166,213)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{72925}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-166,213)$ as 72925, hence both give $AB=\\sqrt{72925}$.",
"robustness_analysis": "Robustness note: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{72925}$.) |
math-001051 | Coordinate Geometry: Distance Formula | 1 | Question: Let $A(-65,96)$ and $B(94,157)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitl... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=94-(-65)=159$ and $\\Delta y=157-(96)=61$.",
"Step 2: A translation sends $... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{29002}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(159,61)$ as 29002, hence both give $AB=\\sqrt{29002}$.",
"robustness_analysis": "Sensitivity analysis: ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{29002}$.) |
math-001052 | Vector Geometry: Norms and Dot Products | 1 | State any required conditions first: Let $A(-60,120)$ and $B(123,36)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=123-(-60)=183$ and $\\Delta y=36-(120)=-84$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{40545}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(183,-84)$ as 40545, hence both give $AB=\\sqrt{40545}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{40545}$.) |
math-001053 | Coordinate Geometry: Distance Formula | 1 | Derive the result step-by-step: Let $A(47,153)$ and $B(141,-177)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=141-(47)=94$ and $\\Delta y=-177-(153)=-330$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{117736}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(94,-330)$ as 117736, hence both give $AB=\\sqrt{117736}$.",
"robustness_analysis": "Generalit... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001054 | Coordinate Geometry: Distance Formula | 1 | Solve (and briefly cross-validate): Let $A(-62,-107)$ and $B(-26,184)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-26-(-62)=36$ and $\\Delta y=184-(-107)=291$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{85977}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(36,291)$ as 85977, hence both give $AB=\\sqrt{85977}$.",
"robustness_analysis": "Generality no... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{85977}$.) |
math-001055 | Vector Geometry: Norms and Dot Products | 1 | Work this out carefully: Let $A(-109,-134)$ and $B(21,-196)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorea... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=21-(-109)=130$ and $\\Delta y=-196-(-134)=-62$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{20744}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(130,-62)$ as 20744, hence both give $AB=\\sqrt{20744}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{20744}$.) |
math-001056 | Analytic Geometry: Translation Invariance | 1 | Give a theorem-based solution: Let $A(-41,-125)$ and $B(-30,-70)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(11,55)$.",
"Step 2: C... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{3146}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(11,55)$ as 3146, hence both give $AB=\\sqrt{3146}$.",
"robustness_analysis": "Sensitivity analy... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{3146}$.) |
math-001057 | Coordinate Geometry: Distance Formula | 1 | Answer using clear logical steps: Let $A(176,-130)$ and $B(-26,58)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-202,188)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{76148}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-202,188)$ as 76148, hence both give $AB=\\sqrt{76148}$.",
"robustness_analysis": "Robustness note: The vector/dot-product met... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{76148}$.) |
math-001058 | Coordinate Geometry: Distance Formula | 1 | Derive the result step-by-step: Let $A(-119,-154)$ and $B(71,102)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(190,256)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{101636}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(190,256)$ as 101636, hence both give $AB=\\sqrt{101636}$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001059 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(-62,-147)$ and $B(56,70)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theo... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(118,217)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{61013}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(118,217)$ as 61013, hence both give $AB=\\sqrt{61013}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-product... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001060 | Geometry: Pythagorean Theorem in Coordinates | 1 | Provide both a computational and a conceptual explanation: Let $A(92,31)$ and $B(47,27)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should referen... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-45,-4)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{2041}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-45,-4)$ as 2041, hence both give $AB=\\sqrt{2041}$.",
"robustness_analysis": "If the problem were perturbed: The vector/dot-pr... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{2041}$.) |
math-001061 | Analytic Geometry: Translation Invariance | 1 | Indicate where a theorem is used: Let $A(-83,-137)$ and $B(56,-76)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(139,61)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{23042}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(139,61)$ as 23042, hence both give $AB=\\sqrt{23042}$.",
"robustness_analysis": "Robustness note: The v... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{23042}$.) |
math-001062 | Coordinate Geometry: Distance Formula | 1 | Indicate where a theorem is used: Let $A(175,180)$ and $B(-48,3)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-223,-177)$.",
"Step ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{81058}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-223,-177)$ as 81058, hence both give $AB=\\sqrt{81058}$.",
"robustness_analysis": "Sensitivity analysi... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001063 | Analytic Geometry: Translation Invariance | 1 | Compute the requested quantity: Let $A(-25,160)$ and $B(189,87)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=189-(-25)=214$ and $\\Delta y=87-(160)=-73$.",
"Step 2: A translation sends... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{51125}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(214,-73)$ as 51125, hence both give $AB=\\sqrt{51125}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{51125}$.) |
math-001064 | Vector Geometry: Norms and Dot Products | 1 | Derive the result step-by-step: Let $A(82,47)$ and $B(73,-48)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagor... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=73-(82)=-9$ and $\\Delta y=-48-(47)=-95$.",
"Step 2: A translation sends $A... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{9106}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-9,-95)$ as 9106, hence both give $AB=\\sqrt{9106}$.",
"robustness_analysis": "If the problem w... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{9106}$.) |
math-001065 | Geometry: Pythagorean Theorem in Coordinates | 1 | Compute the requested quantity: Let $A(-84,-100)$ and $B(37,-132)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(121,-32)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{15665}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(121,-32)$ as 15665, hence both give $AB=\\sqrt{15665}$.",
"robustness_analysis": "Robustness note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001066 | Vector Geometry: Norms and Dot Products | 1 | Do not skip justification steps: Let $A(109,39)$ and $B(84,174)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-25,135)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{18850}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-25,135)$ as 18850, hence both give $AB=\\sqrt{18850}$.",
"robustness_analysis": "Generality note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{18850}$.) |
math-001067 | Coordinate Geometry: Distance Formula | 1 | Compute the requested quantity: Let $A(-57,-89)$ and $B(187,158)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=187-(-57)=244$ and $\\Delta y=158-(-89)=247$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{120545}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(244,247)$ as 120545, hence both give $AB=\\sqrt{120545}$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001068 | Geometry: Pythagorean Theorem in Coordinates | 1 | Show all reasoning: Let $A(-42,-119)$ and $B(-39,-104)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean the... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(3,15)$.",
"Step 2: Co... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{234}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(3,15)$ as 234, hence both give $AB=\\sqrt{234}$.",
"robustness_analysis": "Robustness note: The vector/dot-product method genera... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001069 | Geometry: Pythagorean Theorem in Coordinates | 1 | Give reasoning, not just computation: Let $A(116,-64)$ and $B(70,-192)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-46,-128)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{18500}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-46,-128)$ as 18500, hence both give $AB=\\sqrt{18500}$.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{18500}$.) |
math-001070 | Vector Geometry: Norms and Dot Products | 1 | Question: Let $A(-60,40)$ and $B(55,8)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitly.... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=55-(-60)=115$ and $\\Delta y=8-(40)=-32$.",
"Step 2: A translation sends $A... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{14249}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(115,-32)$ as 14249, hence both give $AB=\\sqrt{14249}$.",
"robustness_analysis": "Generality note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001071 | Coordinate Geometry: Distance Formula | 1 | Proceed methodically: Let $A(-27,-193)$ and $B(115,-9)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean the... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(142,184)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{54020}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(142,184)$ as 54020, hence both give $AB=\\sqrt{54020}$.",
"robustness_analysis": "If the problem were perturbed: The vec... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001072 | Vector Geometry: Norms and Dot Products | 1 | Question: Let $A(-97,-103)$ and $B(180,45)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(277,148)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{98633}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(277,148)$ as 98633, hence both give $AB=\\sqrt{98633}$.",
"robustness_analysis": "Generality note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{98633}$.) |
math-001073 | Geometry: Pythagorean Theorem in Coordinates | 1 | Be explicit about assumptions: Let $A(-144,168)$ and $B(-10,-29)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(134,-197)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{56765}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(134,-197)$ as 56765, hence both give $AB=\\sqrt{56765}$.",
"robustness_analysis": "Robustness note: The vector/dot-produ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{56765}$.) |
math-001074 | Coordinate Geometry: Distance Formula | 1 | Write the solution set clearly: Let $A(-67,-103)$ and $B(95,-26)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(162,77)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{32173}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(162,77)$ as 32173, hence both give $AB=\\sqrt{32173}$.",
"robustness_analysis": "Generality note: The vector/dot-product metho... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{32173}$.) |
math-001075 | Analytic Geometry: Translation Invariance | 1 | Where appropriate, name the theorem you use: Let $A(-185,40)$ and $B(149,169)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=149-(-185)=334$ and $\\Delta y=169-(40)=129$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{128197}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(334,129)$ as 128197, hence both give $AB=\\sqrt{128197}$.",
"robustness_analysis": "Robustness note: The vector/dot-pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{128197}$.) |
math-001076 | Analytic Geometry: Translation Invariance | 1 | Exercise: Let $A(-99,-68)$ and $B(172,181)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=172-(-99)=271$ and $\\Delta y=181-(-68)=249$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{135442}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(271,249)$ as 135442, hence both give $AB=\\sqrt{135442}$.",
"robustness_analysis": "If the problem wer... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001077 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve (and briefly cross-validate): Let $A(136,-30)$ and $B(172,197)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(36,227)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{52825}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(36,227)$ as 52825, hence both give $AB=\\sqrt{52825}$.",
"robustness_analysis": "Robustness note: The v... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{52825}$.) |
math-001078 | Analytic Geometry: Translation Invariance | 1 | Explain what is being counted/optimized: Let $A(-30,-136)$ and $B(-91,172)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-91-(-30)=-61$ and $\\Delta y=172-(-136)=308$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{98585}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-61,308)$ as 98585, hence both give $AB=\\sqrt{98585}$.",
"robustness_analysis": "Robustness note: The vector/dot-product meth... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{98585}$.) |
math-001079 | Geometry: Pythagorean Theorem in Coordinates | 1 | Make each step logically reversible (or explain if not): Let $A(-87,-146)$ and $B(199,-41)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should refe... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(286,105)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{92821}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(286,105)$ as 92821, hence both give $AB=\\sqrt{92821}$.",
"robustness_analysis": "If the probl... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{92821}$.) |
math-001080 | Vector Geometry: Norms and Dot Products | 1 | Explain why your operations are valid: Let $A(-82,191)$ and $B(45,17)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the ... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(127,-174)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{46405}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(127,-174)$ as 46405, hence both give $AB=\\sqrt{46405}$.",
"robustness_analysis": "If the prob... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{46405}$.) |
math-001081 | Vector Geometry: Norms and Dot Products | 1 | Where appropriate, name the theorem you use: Let $A(162,172)$ and $B(12,-37)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors ... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-150,-209)$.",
"Step ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{66181}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-150,-209)$ as 66181, hence both give $AB=\\sqrt{66181}$.",
"robustness_analysis": "Generality note: Th... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{66181}$.) |
math-001082 | Geometry: Pythagorean Theorem in Coordinates | 1 | Explain why your operations are valid: Let $A(198,-81)$ and $B(161,23)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=161-(198)=-37$ and $\\Delta y=23-(-81)=104$.",
"Step 2: A translation sends... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{12185}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-37,104)$ as 12185, hence both give $AB=\\sqrt{12185}$.",
"robustness_analysis": "Generality note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001083 | Analytic Geometry: Translation Invariance | 1 | Start by stating any domain restrictions: Let $A(-149,-162)$ and $B(182,16)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors o... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=182-(-149)=331$ and $\\Delta y=16-(-162)=178$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{141245}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(331,178)$ as 141245, hence both give $AB=\\sqrt{141245}$.",
"robustness_analysis": "If the pr... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{141245}$.) |
math-001084 | Analytic Geometry: Translation Invariance | 1 | Solve (and briefly cross-validate): Let $A(81,-168)$ and $B(103,139)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=103-(81)=22$ and $\\Delta y=139-(-168)=307$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{94733}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(22,307)$ as 94733, hence both give $AB=\\sqrt{94733}$.",
"robustness_analysis": "Robustness no... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{94733}$.) |
math-001085 | Analytic Geometry: Translation Invariance | 1 | Provide both a computational and a conceptual explanation: Let $A(-92,-163)$ and $B(-11,178)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should re... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(81,341)$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{122842}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(81,341)$ as 122842, hence both give $AB=\\sqrt{122842}$.",
"robustness_analysis": "Sensitivit... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001086 | Vector Geometry: Norms and Dot Products | 1 | Proceed methodically: Let $A(-135,187)$ and $B(111,-84)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean th... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=111-(-135)=246$ and $\\Delta y=-84-(187)=-271$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{133957}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(246,-271)$ as 133957, hence both give $AB=\\sqrt{133957}$.",
"robustness_analysis": "Robustness note: The vector/dot-product ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001087 | Analytic Geometry: Translation Invariance | 1 | Work carefully and justify each inference: Let $A(-77,-97)$ and $B(139,151)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors o... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=139-(-77)=216$ and $\\Delta y=151-(-97)=248$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{108160}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(216,248)$ as 108160, hence both give $AB=\\sqrt{108160}$.",
"robustness_analysis": "Generality note: T... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001088 | Analytic Geometry: Translation Invariance | 1 | Work this out carefully: Let $A(166,47)$ and $B(-3,-94)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean th... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-3-(166)=-169$ and $\\Delta y=-94-(47)=-141$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{48442}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-169,-141)$ as 48442, hence both give $AB=\\sqrt{48442}$.",
"robustness_analysis": "If the problem were perturbed: The vector/... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{48442}$.) |
math-001089 | Vector Geometry: Norms and Dot Products | 1 | Be explicit about assumptions: Let $A(-72,92)$ and $B(200,91)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagor... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(272,-1)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{73985}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(272,-1)$ as 73985, hence both give $AB=\\sqrt{73985}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-product ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001090 | Geometry: Pythagorean Theorem in Coordinates | 1 | Exercise: Let $A(-134,-69)$ and $B(68,-182)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explic... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=68-(-134)=202$ and $\\Delta y=-182-(-69)=-113$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{53573}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(202,-113)$ as 53573, hence both give $AB=\\sqrt{53573}$.",
"robustness_analysis": "If the problem were perturbed: The vector/d... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{53573}$.) |
math-001091 | Vector Geometry: Norms and Dot Products | 1 | Give a theorem-based solution: Let $A(-195,119)$ and $B(-22,-114)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(173,-233)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{84218}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(173,-233)$ as 84218, hence both give $AB=\\sqrt{84218}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{84218}$.) |
math-001092 | Vector Geometry: Norms and Dot Products | 1 | Keep the final answer in boxed form: Let $A(77,16)$ and $B(-136,-85)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-136-(77)=-213$ and $\\Delta y=-85-(16)=-101$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{55570}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-213,-101)$ as 55570, hence both give $AB=\\sqrt{55570}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{55570}$.) |
math-001093 | Analytic Geometry: Translation Invariance | 1 | Indicate where a theorem is used: Let $A(-171,145)$ and $B(-166,119)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the P... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-166-(-171)=5$ and $\\Delta y=119-(145)=-26$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{701}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(5,-26)$ as 701, hence both give $AB=\\sqrt{701}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-product m... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{701}$.) |
math-001094 | Coordinate Geometry: Distance Formula | 1 | Complete the analysis: Let $A(-44,181)$ and $B(-145,67)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean th... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-145-(-44)=-101$ and $\\Delta y=67-(181)=-114$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{23197}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-101,-114)$ as 23197, hence both give $AB=\\sqrt{23197}$.",
"robustness_analysis": "Robustness note: The vector/dot-product me... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-001095 | Vector Geometry: Norms and Dot Products | 1 | Solve and include a self-check: Let $A(-44,176)$ and $B(-168,-37)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-168-(-44)=-124$ and $\\Delta y=-37-(176)=-213$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{60745}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-124,-213)$ as 60745, hence both give $AB=\\sqrt{60745}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{60745}$.) |
math-001096 | Vector Geometry: Norms and Dot Products | 1 | Prompt: Let $A(-116,-187)$ and $B(104,-67)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explici... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=104-(-116)=220$ and $\\Delta y=-67-(-187)=120$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{62800}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(220,120)$ as 62800, hence both give $AB=\\sqrt{62800}$.",
"robustness_analysis": "Robustness note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{62800}$.) |
math-001097 | Geometry: Pythagorean Theorem in Coordinates | 1 | Explain why your operations are valid: Let $A(-77,56)$ and $B(51,190)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=51-(-77)=128$ and $\\Delta y=190-(56)=134$.",
"Step 2: A translation sends ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{34340}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(128,134)$ as 34340, hence both give $AB=\\sqrt{34340}$.",
"robustness_analysis": "Sensitivity ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{34340}$.) |
math-001098 | Coordinate Geometry: Distance Formula | 1 | Be explicit about assumptions: Let $A(-57,111)$ and $B(10,-10)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythago... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(67,-121)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{19130}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(67,-121)$ as 19130, hence both give $AB=\\sqrt{19130}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{19130}$.) |
math-001099 | Coordinate Geometry: Distance Formula | 1 | Be explicit about assumptions: Let $A(-113,33)$ and $B(143,-30)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=143-(-113)=256$ and $\\Delta y=-30-(33)=-63$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{69505}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(256,-63)$ as 69505, hence both give $AB=\\sqrt{69505}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{69505}$.) |
math-001100 | Analytic Geometry: Translation Invariance | 1 | Give a fully justified solution: Let $A(145,162)$ and $B(-162,-58)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyt... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-162-(145)=-307$ and $\\Delta y=-58-(162)=-220$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{142649}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-307,-220)$ as 142649, hence both give $AB=\\sqrt{142649}$.",
"robustness_analysis": "If the problem were perturbed: Th... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{142649}$.) |
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