id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-000801 | Vector Geometry: Norms and Dot Products | 1 | Do not skip justification steps: Let $A(135,-74)$ and $B(65,108)$ 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 $\\overrightarrow{AB}=(-70,182)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{38024}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-70,182)$ as 38024, hence both give $AB=\\sqrt{38024}$.",
"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-000802 | Analytic Geometry: Translation Invariance | 1 | Explain why your operations are valid: Let $A(-184,145)$ and $B(57,173)$ 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 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 $\\overrightarrow{AB}=(241,28)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{58865}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(241,28)$ as 58865, hence both give $AB=\\sqrt{58865}$.",
"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... | 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 $\boxed{\sqrt{58865}$.) |
math-000803 | Analytic Geometry: Translation Invariance | 1 | Use two approaches if possible: Let $A(-155,-24)$ and $B(-196,60)$ 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 $\\overrightarrow{AB}=(-41,84)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{8737}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-41,84)$ as 8737, hence both give $AB=\\sqrt{8737}$.",
"robustness_analysis": "Robustness note: The vect... | [
{
"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 $\boxed{\sqrt{8737}$.) |
math-000804 | Geometry: Pythagorean Theorem in Coordinates | 1 | Do not skip justification steps: Let $A(-50,-1)$ and $B(-174,-128)$ 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 $\\overrightarrow{AB}=(-124,-127)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{31505}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-124,-127)$ as 31505, hence both give $AB=\\sqrt{31505}$.",
"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... | 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 $\boxed{\sqrt{31505}$.) |
math-000805 | Coordinate Geometry: Distance Formula | 1 | Provide both a computational and a conceptual explanation: Let $A(-115,141)$ and $B(186,-48)$ 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 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 $\\overrightarrow{AB}=(301,-189)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{126322}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(301,-189)$ as 126322, hence both give $AB=\\sqrt{126322}$.",
"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... | 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 $\boxed{\sqrt{126322}$.) |
math-000806 | Vector Geometry: Norms and Dot Products | 1 | Compute the requested quantity: Let $A(-42,163)$ and $B(-104,-117)$ 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 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 $\\overrightarrow{AB}=(-62,-280)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{82244}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-62,-280)$ as 82244, hence both give $AB=\\sqrt{82244}$.",
"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... | 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-000807 | Coordinate Geometry: Distance Formula | 1 | Provide a rigorous solution: Let $A(157,-2)$ and $B(-59,95)$ 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 $\\overrightarrow{AB}=(-216,97)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{56065}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-216,97)$ as 56065, hence both give $AB=\\sqrt{56065}$.",
"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 $\boxed{\sqrt{56065}$.) |
math-000808 | Analytic Geometry: Translation Invariance | 1 | Task: Let $A(-169,-156)$ and $B(-131,-97)$ 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 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 $\\overrightarrow{AB}=(38,59)$.",
"Step 2: C... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{4925}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(38,59)$ as 4925, hence both give $AB=\\sqrt{4925}$.",
"robustness_analysis": "Generality 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... | 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 $\boxed{\sqrt{4925}$.) |
math-000809 | Geometry: Pythagorean Theorem in Coordinates | 1 | Make each step logically reversible (or explain if not): Let $A(59,12)$ and $B(-36,-34)$ 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": "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: $\\Delta x=-36-(59)=-95$ and $\\Delta y=-34-(12)=-46$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{11141}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-95,-46)$ as 11141, hence both give $AB=\\sqrt{11141}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{11141}$.) |
math-000810 | Vector Geometry: Norms and Dot Products | 1 | Start by stating any domain restrictions: Let $A(150,5)$ and $B(-25,67)$ 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 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 $\\overrightarrow{AB}=(-175,62)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{34469}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-175,62)$ as 34469, hence both give $AB=\\sqrt{34469}$.",
"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... | 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 $\boxed{\sqrt{34469}$.) |
math-000811 | Coordinate Geometry: Distance Formula | 1 | Track quantifiers carefully: Let $A(-134,-117)$ and $B(126,19)$ 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: $\\Delta x=126-(-134)=260$ and $\\Delta y=19-(-117)=136$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{86096}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(260,136)$ as 86096, hence both give $AB=\\sqrt{86096}$.",
"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... | 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 $\boxed{\sqrt{86096}$.) |
math-000812 | Vector Geometry: Norms and Dot Products | 1 | Find the exact value: Let $A(19,194)$ and $B(131,159)$ 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 $\\overrightarrow{AB}=(112,-35)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{13769}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(112,-35)$ as 13769, hence both give $AB=\\sqrt{13769}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000813 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(200,-135)$ and $B(54,178)$ 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 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 $\\overrightarrow{AB}=(-146,313)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{119285}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-146,313)$ as 119285, hence both give $AB=\\sqrt{119285}$.",
"robustness_analysis": "If the problem were perturbed: The vecto... | [
{
"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-000814 | Geometry: Pythagorean Theorem in Coordinates | 1 | Make each step logically reversible (or explain if not): Let $A(-138,189)$ and $B(-77,62)$ 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 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: $\\Delta x=-77-(-138)=61$ and $\\Delta y=62-(189)=-127$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{19850}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(61,-127)$ as 19850, hence both give $AB=\\sqrt{19850}$.",
"robustness_analysis": "If the problem were 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... | 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 $\boxed{\sqrt{19850}$.) |
math-000815 | Geometry: Pythagorean Theorem in Coordinates | 1 | Determine the requested value: Let $A(-136,-64)$ and $B(-69,117)$ 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 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 $\\overrightarrow{AB}=(67,181)$.",
"Step 2: ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{37250}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(67,181)$ as 37250, hence both give $AB=\\sqrt{37250}$.",
"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-000816 | Analytic Geometry: Translation Invariance | 1 | Give an answer and a quick verification: Let $A(-50,36)$ and $B(-164,-66)$ 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 ... | [
{
"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: $\\Delta x=-164-(-50)=-114$ and $\\Delta y=-66-(36)=-102$.",
"Step 2: A translation se... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{23400}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-114,-102)$ as 23400, hence both give $AB=\\sqrt{23400}$.",
"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... | 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-000817 | Vector Geometry: Norms and Dot Products | 1 | Exercise: Let $A(151,31)$ and $B(-28,-55)$ 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 $\\overrightarrow{AB}=(-179,-86)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{39437}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-179,-86)$ as 39437, hence both give $AB=\\sqrt{39437}$.",
"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... | 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-000818 | Vector Geometry: Norms and Dot Products | 1 | Make each step logically reversible (or explain if not): Let $A(-10,-183)$ and $B(73,195)$ 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: $\\Delta x=73-(-10)=83$ and $\\Delta y=195-(-183)=378$.",
"Step 2: A translation sends... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{149773}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(83,378)$ as 149773, hence both give $AB=\\sqrt{149773}$.",
"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... | 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 $\boxed{\sqrt{149773}$.) |
math-000819 | Analytic Geometry: Translation Invariance | 1 | Make each step logically reversible (or explain if not): Let $A(167,-17)$ and $B(-166,2)$ 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": "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 $\\overrightarrow{AB}=(-333,19)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{111250}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-333,19)$ as 111250, hence both give $AB=\\sqrt{111250}$.",
"robustness_analysis": "Robustness 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... | 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 $\boxed{\sqrt{111250}$.) |
math-000820 | Vector Geometry: Norms and Dot Products | 1 | Answer with a short justification: Let $A(78,133)$ and $B(-135,-169)$ 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 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: $\\Delta x=-135-(78)=-213$ and $\\Delta y=-169-(133)=-302$.",
"Step 2: A translation s... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{136573}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-213,-302)$ as 136573, hence both give $AB=\\sqrt{136573}$.",
"robustness_analysis": "If 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 $\boxed{\sqrt{136573}$.) |
math-000821 | Coordinate Geometry: Distance Formula | 1 | Question: Let $A(-109,98)$ and $B(41,78)$ 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 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 $\\overrightarrow{AB}=(150,-20)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{22900}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(150,-20)$ as 22900, hence both give $AB=\\sqrt{22900}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000822 | Analytic Geometry: Translation Invariance | 1 | Explain what is being counted/optimized: Let $A(-149,-117)$ and $B(-100,-34)$ 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: $\\Delta x=-100-(-149)=49$ and $\\Delta y=-34-(-117)=83$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{9290}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(49,83)$ as 9290, hence both give $AB=\\sqrt{9290}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{9290}$.) |
math-000823 | Vector Geometry: Norms and Dot Products | 1 | Exercise: Let $A(-105,-72)$ and $B(127,116)$ 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: $\\Delta x=127-(-105)=232$ and $\\Delta y=116-(-72)=188$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{89168}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(232,188)$ as 89168, hence both give $AB=\\sqrt{89168}$.",
"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... | 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 $\boxed{\sqrt{89168}$.) |
math-000824 | Analytic Geometry: Translation Invariance | 1 | Warm-up: Let $A(112,-139)$ and $B(-162,-127)$ 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 expli... | [
{
"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 $\\overrightarrow{AB}=(-274,12)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{75220}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-274,12)$ as 75220, hence both give $AB=\\sqrt{75220}$.",
"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-000825 | Coordinate Geometry: Distance Formula | 1 | Solve with verification: Let $A(-142,16)$ and $B(-99,-25)$ 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 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 $\\overrightarrow{AB}=(43,-41)$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{3530}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(43,-41)$ as 3530, hence both give $AB=\\sqrt{3530}$.",
"robustness_analysis": "Generality 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... | 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 $\boxed{\sqrt{3530}$.) |
math-000826 | Geometry: Pythagorean Theorem in Coordinates | 1 | Where appropriate, name the theorem you use: Let $A(103,-95)$ and $B(93,-127)$ 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 $\\overrightarrow{AB}=(-10,-32)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{1124}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-10,-32)$ as 1124, hence both give $AB=\\sqrt{1124}$.",
"robustness_analysis": "Robustness note: The vector/dot-product method ... | [
{
"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 $\boxed{\sqrt{1124}$.) |
math-000827 | Vector Geometry: Norms and Dot Products | 1 | Prompt: Let $A(-51,178)$ and $B(128,103)$ 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: $\\Delta x=128-(-51)=179$ and $\\Delta y=103-(178)=-75$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{37666}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(179,-75)$ as 37666, hence both give $AB=\\sqrt{37666}$.",
"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... | 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-000828 | Analytic Geometry: Translation Invariance | 1 | Answer with a short justification: Let $A(-75,37)$ and $B(169,33)$ 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 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: $\\Delta x=169-(-75)=244$ and $\\Delta y=33-(37)=-4$.",
"Step 2: A translation sends $... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{59552}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(244,-4)$ as 59552, hence both give $AB=\\sqrt{59552}$.",
"robustness_analysis": "If the problem were pe... | [
{
"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-000829 | Vector Geometry: Norms and Dot Products | 1 | Answer using clear logical steps: Let $A(-88,-6)$ and $B(-39,162)$ 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 $\\overrightarrow{AB}=(49,168)$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{30625}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(49,168)$ as 30625, hence both give $AB=\\sqrt{30625}$.",
"robustness_analysis": "If the proble... | [
{
"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 $\boxed{\sqrt{30625}$.) |
math-000830 | Geometry: Pythagorean Theorem in Coordinates | 1 | Indicate where a theorem is used: Let $A(77,111)$ and $B(-2,44)$ 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 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: $\\Delta x=-2-(77)=-79$ and $\\Delta y=44-(111)=-67$.",
"Step 2: A translation sends $... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{10730}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-79,-67)$ as 10730, hence both give $AB=\\sqrt{10730}$.",
"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... | 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-000831 | Analytic Geometry: Translation Invariance | 1 | Checkpoint: Let $A(-138,-125)$ and $B(16,-180)$ 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 exp... | [
{
"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 $\\overrightarrow{AB}=(154,-55)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{26741}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(154,-55)$ as 26741, hence both give $AB=\\sqrt{26741}$.",
"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 $\boxed{\sqrt{26741}$.) |
math-000832 | Coordinate Geometry: Distance Formula | 1 | Solve and include a self-check: Let $A(172,-7)$ and $B(-195,-160)$ 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: $\\Delta x=-195-(172)=-367$ and $\\Delta y=-160-(-7)=-153$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{158098}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-367,-153)$ as 158098, hence both give $AB=\\sqrt{158098}$.",
"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... | 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 $\boxed{\sqrt{158098}$.) |
math-000833 | Analytic Geometry: Translation Invariance | 1 | Complete the analysis: Let $A(-84,-34)$ and $B(-47,155)$ 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 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: $\\Delta x=-47-(-84)=37$ and $\\Delta y=155-(-34)=189$.",
"Step 2: A translation sends... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{37090}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(37,189)$ as 37090, hence both give $AB=\\sqrt{37090}$.",
"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... | 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 $\boxed{\sqrt{37090}$.) |
math-000834 | Geometry: Pythagorean Theorem in Coordinates | 1 | Give a theorem-based solution: Let $A(192,-73)$ and $B(120,-3)$ 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 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 $\\overrightarrow{AB}=(-72,70)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{10084}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-72,70)$ as 10084, hence both give $AB=\\sqrt{10084}$.",
"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-000835 | Coordinate Geometry: Distance Formula | 1 | Task: Let $A(-57,-199)$ and $B(41,162)$ 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: $\\Delta x=41-(-57)=98$ and $\\Delta y=162-(-199)=361$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{139925}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(98,361)$ as 139925, hence both give $AB=\\sqrt{139925}$.",
"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... | 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-000836 | Geometry: Pythagorean Theorem in Coordinates | 1 | Explain each transformation: Let $A(-92,164)$ and $B(31,-91)$ 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 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: $\\Delta x=31-(-92)=123$ and $\\Delta y=-91-(164)=-255$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{80154}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(123,-255)$ as 80154, hence both give $AB=\\sqrt{80154}$.",
"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... | 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 $\boxed{\sqrt{80154}$.) |
math-000837 | Analytic Geometry: Translation Invariance | 1 | Give reasoning, not just computation: Let $A(170,-131)$ and $B(58,141)$ 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 $\\overrightarrow{AB}=(-112,272)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{86528}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-112,272)$ as 86528, hence both give $AB=\\sqrt{86528}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000838 | Vector Geometry: Norms and Dot Products | 1 | Provide a rigorous solution: Let $A(-194,74)$ and $B(32,-52)$ 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 Pythagore... | [
{
"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 $\\overrightarrow{AB}=(226,-126)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{66952}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(226,-126)$ as 66952, hence both give $AB=\\sqrt{66952}$.",
"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... | 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-000839 | Vector Geometry: Norms and Dot Products | 1 | Solve and sanity-check: Let $A(188,-10)$ and $B(69,-116)$ 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 $\\overrightarrow{AB}=(-119,-106)$.",
"Step ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{25397}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-119,-106)$ as 25397, hence both give $AB=\\sqrt{25397}$.",
"robustness_analysis": "Sensitivity analysis: 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... | 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 $\boxed{\sqrt{25397}$.) |
math-000840 | Analytic Geometry: Translation Invariance | 1 | Challenge: Let $A(-58,-166)$ and $B(-21,124)$ 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 expli... | [
{
"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: $\\Delta x=-21-(-58)=37$ and $\\Delta y=124-(-166)=290$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{85469}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(37,290)$ as 85469, hence both give $AB=\\sqrt{85469}$.",
"robustness_analysis": "If the problem were perturbed: The vect... | [
{
"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-000841 | Vector Geometry: Norms and Dot Products | 1 | Solve (and briefly cross-validate): Let $A(159,36)$ and $B(-53,-48)$ 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 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 $\\overrightarrow{AB}=(-212,-84)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{52000}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-212,-84)$ as 52000, hence both give $AB=\\sqrt{52000}$.",
"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-000842 | Analytic Geometry: Translation Invariance | 1 | Try to avoid pattern-matching; explain why: Let $A(153,34)$ and $B(-152,-115)$ 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... | [
{
"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: $\\Delta x=-152-(153)=-305$ and $\\Delta y=-115-(34)=-149$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{115226}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-305,-149)$ as 115226, hence both give $AB=\\sqrt{115226}$.",
"robustness_analysis": "Robustness note: 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... | 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-000843 | Geometry: Pythagorean Theorem in Coordinates | 1 | Work this out carefully: Let $A(-194,158)$ and $B(-91,-157)$ 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 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: $\\Delta x=-91-(-194)=103$ and $\\Delta y=-157-(158)=-315$.",
"Step 2: A translation s... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{109834}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(103,-315)$ as 109834, hence both give $AB=\\sqrt{109834}$.",
"robustness_analysis": "If the problem were perturbed: The vecto... | [
{
"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 $\boxed{\sqrt{109834}$.) |
math-000844 | Analytic Geometry: Translation Invariance | 1 | Solve and include a self-check: Let $A(-141,136)$ and $B(-182,-78)$ 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: $\\Delta x=-182-(-141)=-41$ and $\\Delta y=-78-(136)=-214$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{47477}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-41,-214)$ as 47477, hence both give $AB=\\sqrt{47477}$.",
"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... | 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 $\boxed{\sqrt{47477}$.) |
math-000845 | Geometry: Pythagorean Theorem in Coordinates | 1 | Task: Let $A(-37,-39)$ and $B(-96,-190)$ 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 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: $\\Delta x=-96-(-37)=-59$ and $\\Delta y=-190-(-39)=-151$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{26282}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-59,-151)$ as 26282, hence both give $AB=\\sqrt{26282}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000846 | Coordinate Geometry: Distance Formula | 1 | Explain what is being counted/optimized: Let $A(3,-128)$ and $B(-153,189)$ 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: $\\Delta x=-153-(3)=-156$ and $\\Delta y=189-(-128)=317$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{124825}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-156,317)$ as 124825, hence both give $AB=\\sqrt{124825}$.",
"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... | 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-000847 | Analytic Geometry: Translation Invariance | 1 | Indicate where a theorem is used: Let $A(79,182)$ and $B(-30,35)$ 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 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 $\\overrightarrow{AB}=(-109,-147)$.",
"Step ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{33490}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-109,-147)$ as 33490, hence both give $AB=\\sqrt{33490}$.",
"robustness_analysis": "Generality 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-000848 | Vector Geometry: Norms and Dot Products | 1 | Provide both a computational and a conceptual explanation: Let $A(95,-158)$ and $B(142,-185)$ 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 $\\overrightarrow{AB}=(47,-27)$.",
"Step 2: ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{2938}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(47,-27)$ as 2938, hence both give $AB=\\sqrt{2938}$.",
"robustness_analysis": "Sensitivity anal... | [
{
"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 $\boxed{\sqrt{2938}$.) |
math-000849 | Analytic Geometry: Translation Invariance | 1 | Warm-up: Let $A(40,27)$ and $B(-135,168)$ 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: $\\Delta x=-135-(40)=-175$ and $\\Delta y=168-(27)=141$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{50506}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-175,141)$ as 50506, hence both give $AB=\\sqrt{50506}$.",
"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... | 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 $\boxed{\sqrt{50506}$.) |
math-000850 | Geometry: Pythagorean Theorem in Coordinates | 1 | Give reasoning, not just computation: Let $A(46,-183)$ and $B(128,-144)$ 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: $\\Delta x=128-(46)=82$ and $\\Delta y=-144-(-183)=39$.",
"Step 2: A translation sends... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{8245}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(82,39)$ as 8245, hence both give $AB=\\sqrt{8245}$.",
"robustness_analysis": "Sensitivity analysis: 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... | 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-000851 | Vector Geometry: Norms and Dot Products | 1 | Checkpoint: Let $A(-172,-23)$ and $B(-177,71)$ 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": "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 $\\overrightarrow{AB}=(-5,94)$.",
"Step 2: C... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{8861}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-5,94)$ as 8861, hence both give $AB=\\sqrt{8861}$.",
"robustness_analysis": "Generality 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... | 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 $\boxed{\sqrt{8861}$.) |
math-000852 | Coordinate Geometry: Distance Formula | 1 | Where appropriate, name the theorem you use: Let $A(97,176)$ and $B(46,50)$ 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: $\\Delta x=46-(97)=-51$ and $\\Delta y=50-(176)=-126$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{18477}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-51,-126)$ as 18477, hence both give $AB=\\sqrt{18477}$.",
"robustness_analysis": "Generality 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... | 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 $\boxed{\sqrt{18477}$.) |
math-000853 | Vector Geometry: Norms and Dot Products | 1 | Do not skip justification steps: Let $A(-110,50)$ and $B(72,40)$ 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: $\\Delta x=72-(-110)=182$ and $\\Delta y=40-(50)=-10$.",
"Step 2: A translation sends ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{33224}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(182,-10)$ as 33224, hence both give $AB=\\sqrt{33224}$.",
"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 $\boxed{\sqrt{33224}$.) |
math-000854 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve (and briefly cross-validate): Let $A(136,-128)$ and $B(-30,72)$ 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 $\\overrightarrow{AB}=(-166,200)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{67556}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-166,200)$ as 67556, hence both give $AB=\\sqrt{67556}$.",
"robustness_analysis": "If the problem were ... | [
{
"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-000855 | Vector Geometry: Norms and Dot Products | 1 | Track units/moduli carefully: Let $A(179,130)$ and $B(-94,143)$ 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: $\\Delta x=-94-(179)=-273$ and $\\Delta y=143-(130)=13$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{74698}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-273,13)$ as 74698, hence both give $AB=\\sqrt{74698}$.",
"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... | 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-000856 | Vector Geometry: Norms and Dot Products | 1 | Answer with a short justification: Let $A(-138,-196)$ and $B(22,-74)$ 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 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: $\\Delta x=22-(-138)=160$ and $\\Delta y=-74-(-196)=122$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{40484}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(160,122)$ as 40484, hence both give $AB=\\sqrt{40484}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{40484}$.) |
math-000857 | Coordinate Geometry: Distance Formula | 1 | Make each step logically reversible (or explain if not): Let $A(100,160)$ and $B(-76,138)$ 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": "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 $\\overrightarrow{AB}=(-176,-22)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{31460}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-176,-22)$ as 31460, hence both give $AB=\\sqrt{31460}$.",
"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... | 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 $\boxed{\sqrt{31460}$.) |
math-000858 | Coordinate Geometry: Distance Formula | 1 | Make each step logically reversible (or explain if not): Let $A(-179,-92)$ and $B(92,67)$ 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 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: $\\Delta x=92-(-179)=271$ and $\\Delta y=67-(-92)=159$.",
"Step 2: A translation sends... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{98722}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(271,159)$ as 98722, hence both give $AB=\\sqrt{98722}$.",
"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... | 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 $\boxed{\sqrt{98722}$.) |
math-000859 | Analytic Geometry: Translation Invariance | 1 | Try to avoid pattern-matching; explain why: Let $A(-4,91)$ and $B(163,194)$ 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: $\\Delta x=163-(-4)=167$ and $\\Delta y=194-(91)=103$.",
"Step 2: A translation sends ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{38498}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(167,103)$ as 38498, hence both give $AB=\\sqrt{38498}$.",
"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... | 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-000860 | Analytic Geometry: Translation Invariance | 1 | Problem: Let $A(67,-75)$ and $B(-182,-119)$ 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 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: $\\Delta x=-182-(67)=-249$ and $\\Delta y=-119-(-75)=-44$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{63937}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-249,-44)$ as 63937, hence both give $AB=\\sqrt{63937}$.",
"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... | 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-000861 | Vector Geometry: Norms and Dot Products | 1 | Track units/moduli carefully: Let $A(7,-105)$ and $B(-101,68)$ 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 $\\overrightarrow{AB}=(-108,173)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{41593}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-108,173)$ as 41593, hence both give $AB=\\sqrt{41593}$.",
"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-000862 | Vector Geometry: Norms and Dot Products | 1 | Question: Let $A(120,-9)$ and $B(86,-51)$ 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: $\\Delta x=86-(120)=-34$ and $\\Delta y=-51-(-9)=-42$.",
"Step 2: A translation sends ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{2920}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-34,-42)$ as 2920, hence both give $AB=\\sqrt{2920}$.",
"robustness_analysis": "Generality note: 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... | 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-000863 | Vector Geometry: Norms and Dot Products | 1 | Solve (and briefly cross-validate): Let $A(-8,132)$ and $B(101,-46)$ 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 $\\overrightarrow{AB}=(109,-178)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{43565}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(109,-178)$ as 43565, hence both give $AB=\\sqrt{43565}$.",
"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 $\boxed{\sqrt{43565}$.) |
math-000864 | Geometry: Pythagorean Theorem in Coordinates | 1 | Give a fully justified solution: Let $A(-6,101)$ and $B(-84,-136)$ 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 $\\overrightarrow{AB}=(-78,-237)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{62253}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-78,-237)$ as 62253, hence both give $AB=\\sqrt{62253}$.",
"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... | 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-000865 | Geometry: Pythagorean Theorem in Coordinates | 1 | Determine the requested value: Let $A(6,-137)$ and $B(-88,71)$ 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 $\\overrightarrow{AB}=(-94,208)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{52100}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-94,208)$ as 52100, hence both give $AB=\\sqrt{52100}$.",
"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... | 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 $\boxed{\sqrt{52100}$.) |
math-000866 | Vector Geometry: Norms and Dot Products | 1 | Carefully track domains: Let $A(155,160)$ and $B(4,36)$ 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 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 $\\overrightarrow{AB}=(-151,-124)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{38177}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-151,-124)$ as 38177, hence both give $AB=\\sqrt{38177}$.",
"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... | 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 $\boxed{\sqrt{38177}$.) |
math-000867 | Geometry: Pythagorean Theorem in Coordinates | 1 | State any required conditions first: Let $A(81,-75)$ and $B(76,-139)$ 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: $\\Delta x=76-(81)=-5$ and $\\Delta y=-139-(-75)=-64$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{4121}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-5,-64)$ as 4121, hence both give $AB=\\sqrt{4121}$.",
"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... | 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-000868 | Vector Geometry: Norms and Dot Products | 1 | Make each step logically reversible (or explain if not): Let $A(-96,-161)$ and $B(-50,128)$ 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": "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: $\\Delta x=-50-(-96)=46$ and $\\Delta y=128-(-161)=289$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{85637}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(46,289)$ as 85637, hence both give $AB=\\sqrt{85637}$.",
"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-000869 | Analytic Geometry: Translation Invariance | 1 | Solve and justify each step: Let $A(-127,44)$ and $B(161,-57)$ 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 $\\overrightarrow{AB}=(288,-101)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{93145}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(288,-101)$ as 93145, hence both give $AB=\\sqrt{93145}$.",
"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... | 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 $\boxed{\sqrt{93145}$.) |
math-000870 | Vector Geometry: Norms and Dot Products | 1 | Exercise: Let $A(-87,-121)$ and $B(93,-25)$ 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 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 $\\overrightarrow{AB}=(180,96)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{41616}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(180,96)$ as 41616, hence both give $AB=\\sqrt{41616}$.",
"robustness_analysis": "Robustness 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 $\boxed{\sqrt{41616}$.) |
math-000871 | Geometry: Pythagorean Theorem in Coordinates | 1 | Answer with a short justification: Let $A(183,141)$ and $B(134,-147)$ 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 $\\overrightarrow{AB}=(-49,-288)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{85345}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-49,-288)$ as 85345, hence both give $AB=\\sqrt{85345}$.",
"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 $\boxed{\sqrt{85345}$.) |
math-000872 | Analytic Geometry: Translation Invariance | 1 | Try to avoid pattern-matching; explain why: Let $A(-72,120)$ and $B(158,-138)$ 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 $\\overrightarrow{AB}=(230,-258)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{119464}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(230,-258)$ as 119464, hence both give $AB=\\sqrt{119464}$.",
"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... | 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 $\boxed{\sqrt{119464}$.) |
math-000873 | Analytic Geometry: Translation Invariance | 1 | Exercise: Let $A(-64,164)$ and $B(-71,68)$ 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": "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: $\\Delta x=-71-(-64)=-7$ and $\\Delta y=68-(164)=-96$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{9265}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-7,-96)$ as 9265, hence both give $AB=\\sqrt{9265}$.",
"robustness_analysis": "Sensitivity analysis: 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 $\boxed{\sqrt{9265}$.) |
math-000874 | Analytic Geometry: Translation Invariance | 1 | Explain what is being counted/optimized: Let $A(-123,-136)$ and $B(-187,54)$ 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: $\\Delta x=-187-(-123)=-64$ and $\\Delta y=54-(-136)=190$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{40196}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-64,190)$ as 40196, hence both give $AB=\\sqrt{40196}$.",
"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... | 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 $\boxed{\sqrt{40196}$.) |
math-000875 | Coordinate Geometry: Distance Formula | 1 | Task: Let $A(105,119)$ and $B(-153,62)$ 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 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: $\\Delta x=-153-(105)=-258$ and $\\Delta y=62-(119)=-57$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{69813}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-258,-57)$ as 69813, hence both give $AB=\\sqrt{69813}$.",
"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-000876 | Geometry: Pythagorean Theorem in Coordinates | 1 | Challenge: Let $A(-193,-31)$ and $B(190,-132)$ 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 Pythagorean theorem expl... | [
{
"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 $\\overrightarrow{AB}=(383,-101)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{156890}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(383,-101)$ as 156890, hence both give $AB=\\sqrt{156890}$.",
"robustness_analysis": "Generality 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... | 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-000877 | Geometry: Pythagorean Theorem in Coordinates | 1 | Carefully track domains: Let $A(-63,151)$ and $B(-163,199)$ 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... | [
{
"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 $\\overrightarrow{AB}=(-100,48)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{12304}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-100,48)$ as 12304, hence both give $AB=\\sqrt{12304}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{12304}$.) |
math-000878 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve and then verify: Let $A(64,-82)$ and $B(195,-101)$ 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: $\\Delta x=195-(64)=131$ and $\\Delta y=-101-(-82)=-19$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{17522}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(131,-19)$ as 17522, hence both give $AB=\\sqrt{17522}$.",
"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-000879 | Geometry: Pythagorean Theorem in Coordinates | 1 | Exercise: Let $A(190,-149)$ and $B(74,-77)$ 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: $\\Delta x=74-(190)=-116$ and $\\Delta y=-77-(-149)=72$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{18640}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-116,72)$ as 18640, hence both give $AB=\\sqrt{18640}$.",
"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 $\boxed{\sqrt{18640}$.) |
math-000880 | Analytic Geometry: Translation Invariance | 1 | Solve and then verify: Let $A(-136,167)$ and $B(-129,193)$ 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 $\\overrightarrow{AB}=(7,26)$.",
"Step 2: Co... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{725}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(7,26)$ as 725, hence both give $AB=\\sqrt{725}$.",
"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... | 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 $\boxed{\sqrt{725}$.) |
math-000881 | Vector Geometry: Norms and Dot Products | 1 | Track units/moduli carefully: Let $A(38,-9)$ and $B(28,-126)$ 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": "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 $\\overrightarrow{AB}=(-10,-117)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{13789}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-10,-117)$ as 13789, hence both give $AB=\\sqrt{13789}$.",
"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... | 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-000882 | Geometry: Pythagorean Theorem in Coordinates | 1 | Track units/moduli carefully: Let $A(19,111)$ and $B(3,-3)$ 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 Pythagorean... | [
{
"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: $\\Delta x=3-(19)=-16$ and $\\Delta y=-3-(111)=-114$.",
"Step 2: A translation sends $... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{13252}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-16,-114)$ as 13252, hence both give $AB=\\sqrt{13252}$.",
"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... | 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 $\boxed{\sqrt{13252}$.) |
math-000883 | Vector Geometry: Norms and Dot Products | 1 | Start by stating any domain restrictions: Let $A(-175,-64)$ and $B(20,-161)$ 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: $\\Delta x=20-(-175)=195$ and $\\Delta y=-161-(-64)=-97$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{47434}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(195,-97)$ as 47434, hence both give $AB=\\sqrt{47434}$.",
"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... | 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 $\boxed{\sqrt{47434}$.) |
math-000884 | Geometry: Pythagorean Theorem in Coordinates | 1 | Provide a rigorous solution: Let $A(-111,-141)$ and $B(13,176)$ 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: $\\Delta x=13-(-111)=124$ and $\\Delta y=176-(-141)=317$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{115865}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(124,317)$ as 115865, hence both give $AB=\\sqrt{115865}$.",
"robustness_analysis": "Robustness 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... | 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 $\boxed{\sqrt{115865}$.) |
math-000885 | Vector Geometry: Norms and Dot Products | 1 | Solve and justify each step: Let $A(66,-119)$ and $B(-26,180)$ 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: $\\Delta x=-26-(66)=-92$ and $\\Delta y=180-(-119)=299$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{97865}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-92,299)$ as 97865, hence both give $AB=\\sqrt{97865}$.",
"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... | 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 $\boxed{\sqrt{97865}$.) |
math-000886 | Vector Geometry: Norms and Dot Products | 1 | Explain why your operations are valid: Let $A(70,179)$ and $B(13,-145)$ 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: $\\Delta x=13-(70)=-57$ and $\\Delta y=-145-(179)=-324$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{108225}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-57,-324)$ as 108225, hence both give $AB=\\sqrt{108225}$.",
"robustness_analysis": "Generality 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-000887 | Coordinate Geometry: Distance Formula | 1 | Checkpoint: Let $A(-47,-3)$ and $B(-105,-139)$ 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 Pythagorean theorem expl... | [
{
"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 $\\overrightarrow{AB}=(-58,-136)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{21860}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-58,-136)$ as 21860, hence both give $AB=\\sqrt{21860}$.",
"robustness_analysis": "If the problem were ... | [
{
"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 $\boxed{\sqrt{21860}$.) |
math-000888 | Vector Geometry: Norms and Dot Products | 1 | Try to avoid pattern-matching; explain why: Let $A(-21,-128)$ and $B(122,-72)$ 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: $\\Delta x=122-(-21)=143$ and $\\Delta y=-72-(-128)=56$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{23585}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(143,56)$ as 23585, hence both give $AB=\\sqrt{23585}$.",
"robustness_analysis": "If the problem were perturbed: The vect... | [
{
"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 $\boxed{\sqrt{23585}$.) |
math-000889 | Vector Geometry: Norms and Dot Products | 1 | Explain why your operations are valid: Let $A(-200,-63)$ and $B(-174,150)$ 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 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: $\\Delta x=-174-(-200)=26$ and $\\Delta y=150-(-63)=213$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{46045}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(26,213)$ as 46045, hence both give $AB=\\sqrt{46045}$.",
"robustness_analysis": "If the problem were perturbed: The vect... | [
{
"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 $\boxed{\sqrt{46045}$.) |
math-000890 | Vector Geometry: Norms and Dot Products | 1 | Solve (and briefly cross-validate): Let $A(92,-160)$ and $B(-47,112)$ 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: $\\Delta x=-47-(92)=-139$ and $\\Delta y=112-(-160)=272$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{93305}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-139,272)$ as 93305, hence both give $AB=\\sqrt{93305}$.",
"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... | 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-000891 | Geometry: Pythagorean Theorem in Coordinates | 1 | Where appropriate, name the theorem you use: Let $A(189,-121)$ and $B(61,-25)$ 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": "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 $\\overrightarrow{AB}=(-128,96)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{25600}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-128,96)$ as 25600, hence both give $AB=\\sqrt{25600}$.",
"robustness_analysis": "Generality 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... | 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-000892 | Analytic Geometry: Translation Invariance | 1 | Try to avoid pattern-matching; explain why: Let $A(117,44)$ and $B(8,77)$ 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 $\\overrightarrow{AB}=(-109,33)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{12970}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-109,33)$ as 12970, hence both give $AB=\\sqrt{12970}$.",
"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... | 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 $\boxed{\sqrt{12970}$.) |
math-000893 | Vector Geometry: Norms and Dot Products | 1 | Solve and justify each step: Let $A(52,178)$ and $B(42,71)$ 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 $\\overrightarrow{AB}=(-10,-107)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{11549}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-10,-107)$ as 11549, hence both give $AB=\\sqrt{11549}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000894 | Analytic Geometry: Translation Invariance | 1 | Task: Let $A(147,-62)$ and $B(-19,-15)$ 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: $\\Delta x=-19-(147)=-166$ and $\\Delta y=-15-(-62)=47$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{29765}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-166,47)$ as 29765, hence both give $AB=\\sqrt{29765}$.",
"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-000895 | Analytic Geometry: Translation Invariance | 1 | Answer with a short justification: Let $A(-137,15)$ and $B(-161,142)$ 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: $\\Delta x=-161-(-137)=-24$ and $\\Delta y=142-(15)=127$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{16705}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-24,127)$ as 16705, hence both give $AB=\\sqrt{16705}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{16705}$.) |
math-000896 | Analytic Geometry: Translation Invariance | 1 | Complete the analysis: Let $A(126,69)$ and $B(-170,-18)$ 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: $\\Delta x=-170-(126)=-296$ and $\\Delta y=-18-(69)=-87$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{95185}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-296,-87)$ as 95185, hence both give $AB=\\sqrt{95185}$.",
"robustness_analysis": "If the problem were ... | [
{
"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-000897 | Geometry: Pythagorean Theorem in Coordinates | 1 | Proceed methodically: Let $A(40,-16)$ and $B(-152,141)$ 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 $\\overrightarrow{AB}=(-192,157)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{61513}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-192,157)$ as 61513, hence both give $AB=\\sqrt{61513}$.",
"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... | 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-000898 | Analytic Geometry: Translation Invariance | 1 | Proceed methodically: Let $A(200,135)$ and $B(-172,-2)$ 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 $\\overrightarrow{AB}=(-372,-137)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{157153}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-372,-137)$ as 157153, hence both give $AB=\\sqrt{157153}$.",
"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... | 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-000899 | Analytic Geometry: Translation Invariance | 1 | Solve with verification: Let $A(16,187)$ and $B(154,196)$ 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 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: $\\Delta x=154-(16)=138$ and $\\Delta y=196-(187)=9$.",
"Step 2: A translation sends $... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{19125}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(138,9)$ as 19125, hence both give $AB=\\sqrt{19125}$.",
"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... | 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-000900 | Analytic Geometry: Translation Invariance | 1 | Work this out carefully: Let $A(152,-86)$ and $B(36,-167)$ 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 $\\overrightarrow{AB}=(-116,-81)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{20017}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-116,-81)$ as 20017, hence both give $AB=\\sqrt{20017}$.",
"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... | 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 $\boxed{\sqrt{20017}$.) |
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