id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-000901 | Analytic Geometry: Translation Invariance | 1 | Give a theorem-based solution: Let $A(137,-128)$ and $B(13,118)$ 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: $\\Delta x=13-(137)=-124$ and $\\Delta y=118-(-128)=246$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{75892}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-124,246)$ as 75892, hence both give $AB=\\sqrt{75892}$.",
"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... | 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 $\boxed{\sqrt{75892}$.) |
math-000902 | Analytic Geometry: Translation Invariance | 1 | Determine the requested value: Let $A(152,77)$ and $B(-53,147)$ 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 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: $\\Delta x=-53-(152)=-205$ and $\\Delta y=147-(77)=70$.",
"Step 2: A translation sends... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{46925}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-205,70)$ as 46925, hence both give $AB=\\sqrt{46925}$.",
"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... | 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 $\boxed{\sqrt{46925}$.) |
math-000903 | Geometry: Pythagorean Theorem in Coordinates | 1 | Derive the result step-by-step: Let $A(3,68)$ and $B(-86,37)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-89,-31)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{8882}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-89,-31)$ as 8882, hence both give $AB=\\sqrt{8882}$.",
"robustness_analysis": "Sensitivity ana... | [
{
"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-000904 | Vector Geometry: Norms and Dot Products | 1 | Find the exact value: Let $A(-189,-117)$ and $B(-110,153)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean ... | [
{
"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 $\\overrightarrow{AB}=(79,270)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{79141}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(79,270)$ as 79141, hence both give $AB=\\sqrt{79141}$.",
"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-000905 | Coordinate Geometry: Distance Formula | 1 | Write the solution set clearly: Let $A(80,56)$ and $B(-180,91)$ 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 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: $\\Delta x=-180-(80)=-260$ and $\\Delta y=91-(56)=35$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{68825}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-260,35)$ as 68825, hence both give $AB=\\sqrt{68825}$.",
"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... | 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 $\boxed{\sqrt{68825}$.) |
math-000906 | Coordinate Geometry: Distance Formula | 1 | Proceed methodically: Let $A(91,180)$ and $B(50,4)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=50-(91)=-41$ and $\\Delta y=4-(180)=-176$.",
"Step 2: A translation sends $... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{32657}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-41,-176)$ as 32657, hence both give $AB=\\sqrt{32657}$.",
"robustness_analysis": "Robustness note: The vector/dot-produ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000907 | Coordinate Geometry: Distance Formula | 1 | Solve (and briefly cross-validate): Let $A(-73,-19)$ and $B(132,175)$ 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 $\\overrightarrow{AB}=(205,194)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{79661}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(205,194)$ as 79661, hence both give $AB=\\sqrt{79661}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000908 | Geometry: Pythagorean Theorem in Coordinates | 1 | Keep the final answer in boxed form: Let $A(87,117)$ and $B(-13,-60)$ 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 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: $\\Delta x=-13-(87)=-100$ and $\\Delta y=-60-(117)=-177$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{41329}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-100,-177)$ as 41329, hence both give $AB=\\sqrt{41329}$.",
"robustness_analysis": "If the pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{41329}$.) |
math-000909 | Vector Geometry: Norms and Dot Products | 1 | State any required conditions first: Let $A(-16,-88)$ and $B(-13,-149)$ 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 $\\overrightarrow{AB}=(3,-61)$.",
"Step 2: C... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{3730}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(3,-61)$ as 3730, hence both give $AB=\\sqrt{3730}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{3730}$.) |
math-000910 | Analytic Geometry: Translation Invariance | 1 | Solve and justify each step: Let $A(154,176)$ and $B(-109,-8)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-263,-184)$.",
"Step ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{103025}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-263,-184)$ as 103025, hence both give $AB=\\sqrt{103025}$.",
"robustness_analysis": "General... | [
{
"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 $\boxed{\sqrt{103025}$.) |
math-000911 | Vector Geometry: Norms and Dot Products | 1 | Answer with a short justification: Let $A(120,166)$ and $B(108,48)$ 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 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: $\\Delta x=108-(120)=-12$ and $\\Delta y=48-(166)=-118$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{14068}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-12,-118)$ as 14068, hence both give $AB=\\sqrt{14068}$.",
"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-000912 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(101,152)$ and $B(-72,30)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-173,-122)$.",
"Step ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{44813}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-173,-122)$ as 44813, hence both give $AB=\\sqrt{44813}$.",
"robustness_analysis": "Generality... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{44813}$.) |
math-000913 | Vector Geometry: Norms and Dot Products | 1 | Explain what is being counted/optimized: Let $A(-112,22)$ and $B(-179,-177)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors o... | [
{
"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 $\\overrightarrow{AB}=(-67,-199)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{44090}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-67,-199)$ as 44090, hence both give $AB=\\sqrt{44090}$.",
"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-000914 | Analytic Geometry: Translation Invariance | 1 | Give a theorem-based solution: Let $A(146,149)$ and $B(-178,23)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-178-(146)=-324$ and $\\Delta y=23-(149)=-126$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{120852}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-324,-126)$ as 120852, hence both give $AB=\\sqrt{120852}$.",
"robustness_analysis": "Sensitivity analysis: The vector/... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000915 | Analytic Geometry: Translation Invariance | 1 | Use two approaches if possible: Let $A(-17,-77)$ and $B(-168,125)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-168-(-17)=-151$ and $\\Delta y=125-(-77)=202$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{63605}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-151,202)$ as 63605, hence both give $AB=\\sqrt{63605}$.",
"robustness_analysis": "Robustness note: The vector/dot-product met... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{63605}$.) |
math-000916 | Coordinate Geometry: Distance Formula | 1 | Checkpoint: Let $A(-172,-95)$ and $B(-17,142)$ 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 Pythagorean theorem expl... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-17-(-172)=155$ and $\\Delta y=142-(-95)=237$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{80194}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(155,237)$ as 80194, hence both give $AB=\\sqrt{80194}$.",
"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-000917 | Coordinate Geometry: Distance Formula | 1 | Solve and justify each step: Let $A(94,-66)$ and $B(34,67)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean... | [
{
"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 $\\overrightarrow{AB}=(-60,133)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{21289}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-60,133)$ as 21289, hence both give $AB=\\sqrt{21289}$.",
"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 $\boxed{\sqrt{21289}$.) |
math-000918 | Geometry: Pythagorean Theorem in Coordinates | 1 | Complete the analysis: Let $A(-9,-89)$ and $B(-115,14)$ 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": "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: $\\Delta x=-115-(-9)=-106$ and $\\Delta y=14-(-89)=103$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{21845}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-106,103)$ as 21845, hence both give $AB=\\sqrt{21845}$.",
"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... | 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 $\boxed{\sqrt{21845}$.) |
math-000919 | Analytic Geometry: Translation Invariance | 1 | Explain why your operations are valid: Let $A(160,83)$ and $B(-11,-196)$ 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 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: $\\Delta x=-11-(160)=-171$ and $\\Delta y=-196-(83)=-279$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{107082}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-171,-279)$ as 107082, hence both give $AB=\\sqrt{107082}$.",
"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... | 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-000920 | Coordinate Geometry: Distance Formula | 1 | Compute the requested quantity: Let $A(20,-192)$ and $B(-75,-107)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or 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: $\\Delta x=-75-(20)=-95$ and $\\Delta y=-107-(-192)=85$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{16250}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-95,85)$ as 16250, hence both give $AB=\\sqrt{16250}$.",
"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 $\boxed{\sqrt{16250}$.) |
math-000921 | Analytic Geometry: Translation Invariance | 1 | Answer using clear logical steps: Let $A(25,80)$ and $B(-4,-93)$ 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: $\\Delta x=-4-(25)=-29$ and $\\Delta y=-93-(80)=-173$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{30770}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-29,-173)$ as 30770, hence both give $AB=\\sqrt{30770}$.",
"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... | 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-000922 | Analytic Geometry: Translation Invariance | 1 | Track units/moduli carefully: Let $A(-184,-104)$ and $B(33,-5)$ 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 $\\overrightarrow{AB}=(217,99)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{56890}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(217,99)$ as 56890, hence both give $AB=\\sqrt{56890}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-product ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000923 | Vector Geometry: Norms and Dot Products | 1 | Compute the requested quantity: Let $A(70,137)$ and $B(-26,114)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-96,-23)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{9745}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-96,-23)$ as 9745, hence both give $AB=\\sqrt{9745}$.",
"robustness_analysis": "Sensitivity analysis: 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 $\boxed{\sqrt{9745}$.) |
math-000924 | Coordinate Geometry: Distance Formula | 1 | Provide a rigorous solution: Let $A(-99,-87)$ and $B(-59,179)$ 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: $\\Delta x=-59-(-99)=40$ and $\\Delta y=179-(-87)=266$.",
"Step 2: A translation sends... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{72356}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(40,266)$ as 72356, hence both give $AB=\\sqrt{72356}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000925 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(24,139)$ and $B(82,78)$ 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 theore... | [
{
"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: $\\Delta x=82-(24)=58$ and $\\Delta y=78-(139)=-61$.",
"Step 2: A translation sends $A... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{7085}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(58,-61)$ as 7085, hence both give $AB=\\sqrt{7085}$.",
"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... | 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-000926 | Analytic Geometry: Translation Invariance | 1 | Do not skip justification steps: Let $A(109,-163)$ and $B(19,71)$ 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 Pytha... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=19-(109)=-90$ and $\\Delta y=71-(-163)=234$.",
"Step 2: A translation sends... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{62856}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-90,234)$ as 62856, hence both give $AB=\\sqrt{62856}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{62856}$.) |
math-000927 | Vector Geometry: Norms and Dot Products | 1 | Give a fully justified solution: Let $A(69,143)$ and $B(67,134)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-2,-9)$.",
"Step 2: C... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{85}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-2,-9)$ as 85, hence both give $AB=\\sqrt{85}$.",
"robustness_analysis": "If the problem were perturbed: 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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000928 | Coordinate Geometry: Distance Formula | 1 | Explain each transformation: Let $A(-102,-174)$ and $B(55,-32)$ 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: $\\Delta x=55-(-102)=157$ and $\\Delta y=-32-(-174)=142$.",
"Step 2: A translation sen... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{44813}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(157,142)$ as 44813, hence both give $AB=\\sqrt{44813}$.",
"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 $\boxed{\sqrt{44813}$.) |
math-000929 | Geometry: Pythagorean Theorem in Coordinates | 1 | Explain why your operations are valid: Let $A(138,-92)$ and $B(199,69)$ 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 $\\overrightarrow{AB}=(61,161)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{29642}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(61,161)$ as 29642, hence both give $AB=\\sqrt{29642}$.",
"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 $\boxed{\sqrt{29642}$.) |
math-000930 | Coordinate Geometry: Distance Formula | 1 | Use two approaches if possible: Let $A(96,-15)$ and $B(-93,45)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-93-(96)=-189$ and $\\Delta y=45-(-15)=60$.",
"Step 2: A translation sends ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{39321}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-189,60)$ as 39321, hence both give $AB=\\sqrt{39321}$.",
"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 $\boxed{\sqrt{39321}$.) |
math-000931 | Coordinate Geometry: Distance Formula | 1 | Prompt: Let $A(141,-120)$ and $B(169,146)$ 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: $\\Delta x=169-(141)=28$ and $\\Delta y=146-(-120)=266$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{71540}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(28,266)$ as 71540, hence both give $AB=\\sqrt{71540}$.",
"robustness_analysis": "Generality note: The vector/dot-product metho... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000932 | Vector Geometry: Norms and Dot Products | 1 | Task: Let $A(-85,90)$ and $B(-200,177)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-200-(-85)=-115$ and $\\Delta y=177-(90)=87$.",
"Step 2: A translation send... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{20794}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-115,87)$ as 20794, hence both give $AB=\\sqrt{20794}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000933 | Coordinate Geometry: Distance Formula | 1 | Compute the requested quantity: Let $A(-65,-29)$ and $B(111,109)$ 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 $\\overrightarrow{AB}=(176,138)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{50020}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(176,138)$ as 50020, hence both give $AB=\\sqrt{50020}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000934 | Coordinate Geometry: Distance Formula | 1 | Make each step logically reversible (or explain if not): Let $A(-5,-58)$ and $B(-93,-125)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should 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: $\\Delta x=-93-(-5)=-88$ and $\\Delta y=-125-(-58)=-67$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{12233}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-88,-67)$ as 12233, hence both give $AB=\\sqrt{12233}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000935 | Vector Geometry: Norms and Dot Products | 1 | Track units/moduli carefully: Let $A(56,99)$ and $B(20,24)$ 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: $\\Delta x=20-(56)=-36$ and $\\Delta y=24-(99)=-75$.",
"Step 2: A translation sends $A... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{6921}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-36,-75)$ as 6921, hence both give $AB=\\sqrt{6921}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{6921}$.) |
math-000936 | Analytic Geometry: Translation Invariance | 1 | Solve and then verify: Let $A(-51,123)$ and $B(180,-140)$ 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 $\\overrightarrow{AB}=(231,-263)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{122530}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(231,-263)$ as 122530, hence both give $AB=\\sqrt{122530}$.",
"robustness_analysis": "If the 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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{122530}$.) |
math-000937 | Coordinate Geometry: Distance Formula | 1 | Solve and justify each step: Let $A(-73,192)$ and $B(68,-75)$ 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 $\\overrightarrow{AB}=(141,-267)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{91170}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(141,-267)$ as 91170, hence both give $AB=\\sqrt{91170}$.",
"robustness_analysis": "Robustness note: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000938 | Geometry: Pythagorean Theorem in Coordinates | 1 | Answer using clear logical steps: Let $A(102,30)$ and $B(-198,-108)$ 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 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 $\\overrightarrow{AB}=(-300,-138)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{109044}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-300,-138)$ as 109044, hence both give $AB=\\sqrt{109044}$.",
"robustness_analysis": "Sensitivity analysis: The vector/... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000939 | Analytic Geometry: Translation Invariance | 1 | Solve with verification: Let $A(74,153)$ and $B(-170,-56)$ 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: $\\Delta x=-170-(74)=-244$ and $\\Delta y=-56-(153)=-209$.",
"Step 2: A translation se... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{103217}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-244,-209)$ as 103217, hence both give $AB=\\sqrt{103217}$.",
"robustness_analysis": "Robustn... | [
{
"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 $\boxed{\sqrt{103217}$.) |
math-000940 | Geometry: Pythagorean Theorem in Coordinates | 1 | Warm-up: Let $A(-90,56)$ and $B(-168,-39)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem 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 $\\overrightarrow{AB}=(-78,-95)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{15109}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-78,-95)$ as 15109, hence both give $AB=\\sqrt{15109}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Key idea: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{15109}$.) |
math-000941 | Analytic Geometry: Translation Invariance | 1 | Indicate where a theorem is used: Let $A(-43,-14)$ and $B(135,-40)$ 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 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 $\\overrightarrow{AB}=(178,-26)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{32360}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(178,-26)$ as 32360, hence both give $AB=\\sqrt{32360}$.",
"robustness_analysis": "Sensitivity analysis:... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000942 | Coordinate Geometry: Distance Formula | 1 | Keep the final answer in boxed form: Let $A(22,68)$ and $B(-106,80)$ 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 $\\overrightarrow{AB}=(-128,12)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{16528}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-128,12)$ as 16528, hence both give $AB=\\sqrt{16528}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{16528}$.) |
math-000943 | Vector Geometry: Norms and Dot Products | 1 | Keep the final answer in boxed form: Let $A(-166,45)$ and $B(11,-62)$ 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 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 $\\overrightarrow{AB}=(177,-107)$.",
"Step 2... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{42778}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(177,-107)$ as 42778, hence both give $AB=\\sqrt{42778}$.",
"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 $\boxed{\sqrt{42778}$.) |
math-000944 | Geometry: Pythagorean Theorem in Coordinates | 1 | Where appropriate, name the theorem you use: Let $A(-192,156)$ and $B(-188,-110)$ 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 vect... | [
{
"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: $\\Delta x=-188-(-192)=4$ and $\\Delta y=-110-(156)=-266$.",
"Step 2: A translation se... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{70772}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(4,-266)$ as 70772, hence both give $AB=\\sqrt{70772}$.",
"robustness_analysis": "Generality note: The vector/dot-product metho... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000945 | Coordinate Geometry: Distance Formula | 1 | Show all reasoning: Let $A(-189,-118)$ and $B(55,186)$ 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": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=55-(-189)=244$ and $\\Delta y=186-(-118)=304$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{151952}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(244,304)$ as 151952, hence both give $AB=\\sqrt{151952}$.",
"robustness_analysis": "Robustness note: The vector/dot-product m... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{151952}$.) |
math-000946 | Analytic Geometry: Translation Invariance | 1 | Proceed methodically: Let $A(126,169)$ and $B(137,-174)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean 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 $\\overrightarrow{AB}=(11,-343)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{117770}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(11,-343)$ as 117770, hence both give $AB=\\sqrt{117770}$.",
"robustness_analysis": "Robustness note: The vector/dot-pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000947 | Geometry: Pythagorean Theorem in Coordinates | 1 | Provide both a computational and a conceptual explanation: Let $A(7,126)$ and $B(144,95)$ 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 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 $\\overrightarrow{AB}=(137,-31)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{19730}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(137,-31)$ as 19730, hence both give $AB=\\sqrt{19730}$.",
"robustness_analysis": "Generality note: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Takeaway: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{19730}$.) |
math-000948 | Vector Geometry: Norms and Dot Products | 1 | Solve and sanity-check: Let $A(189,72)$ and $B(-185,-156)$ 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: $\\Delta x=-185-(189)=-374$ and $\\Delta y=-156-(72)=-228$.",
"Step 2: A translation s... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{191860}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-374,-228)$ as 191860, hence both give $AB=\\sqrt{191860}$.",
"robustness_analysis": "If the problem were perturbed: Th... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000949 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(-36,55)$ and $B(-105,46)$ 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 Pythagorean theo... | [
{
"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: $\\Delta x=-105-(-36)=-69$ and $\\Delta y=46-(55)=-9$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{4842}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-69,-9)$ as 4842, hence both give $AB=\\sqrt{4842}$.",
"robustness_analysis": "Generality note: The vector/dot-product method g... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{4842}$.) |
math-000950 | Geometry: Pythagorean Theorem in Coordinates | 1 | Write the solution set clearly: Let $A(93,-69)$ and $B(116,170)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(23,239)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{57650}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(23,239)$ as 57650, hence both give $AB=\\sqrt{57650}$.",
"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-000951 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve and include a self-check: Let $A(26,146)$ and $B(-191,79)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-217,-67)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{51578}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-217,-67)$ as 51578, hence both give $AB=\\sqrt{51578}$.",
"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... | 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 $\boxed{\sqrt{51578}$.) |
math-000952 | Vector Geometry: Norms and Dot Products | 1 | Keep the final answer in boxed form: Let $A(44,75)$ and $B(21,68)$ 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 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 $\\overrightarrow{AB}=(-23,-7)$.",
"Step 2: ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{578}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-23,-7)$ as 578, hence both give $AB=\\sqrt{578}$.",
"robustness_analysis": "Generality note: The vector/dot-product method gene... | [
{
"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 $\boxed{\sqrt{578}$.) |
math-000953 | Vector Geometry: Norms and Dot Products | 1 | Where appropriate, name the theorem you use: Let $A(103,42)$ and $B(139,5)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(36,-37)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{2665}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(36,-37)$ as 2665, hence both give $AB=\\sqrt{2665}$.",
"robustness_analysis": "If the problem were pertu... | [
{
"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 $\boxed{\sqrt{2665}$.) |
math-000954 | Geometry: Pythagorean Theorem in Coordinates | 1 | Do not skip justification steps: Let $A(28,-71)$ and $B(-196,-40)$ 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 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 $\\overrightarrow{AB}=(-224,31)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{51137}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-224,31)$ as 51137, hence both give $AB=\\sqrt{51137}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000955 | Vector Geometry: Norms and Dot Products | 1 | Give a fully justified solution: Let $A(145,-60)$ and $B(74,-102)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pyth... | [
{
"method_name": "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: $\\Delta x=74-(145)=-71$ and $\\Delta y=-102-(-60)=-42$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{6805}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-71,-42)$ as 6805, hence both give $AB=\\sqrt{6805}$.",
"robustness_analysis": "Sensitivity analysis: 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... | 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-000956 | Analytic Geometry: Translation Invariance | 1 | Work this out carefully: Let $A(152,-18)$ and $B(185,-53)$ 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: $\\Delta x=185-(152)=33$ and $\\Delta y=-53-(-18)=-35$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{2314}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(33,-35)$ as 2314, hence both give $AB=\\sqrt{2314}$.",
"robustness_analysis": "If the problem w... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{2314}$.) |
math-000957 | Analytic Geometry: Translation Invariance | 1 | Solve and justify each step: Let $A(52,-134)$ and $B(21,57)$ 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 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 $\\overrightarrow{AB}=(-31,191)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{37442}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-31,191)$ as 37442, hence both give $AB=\\sqrt{37442}$.",
"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... | 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-000958 | Vector Geometry: Norms and Dot Products | 1 | Proceed methodically: Let $A(-32,-199)$ and $B(91,78)$ 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 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 $\\overrightarrow{AB}=(123,277)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{91858}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(123,277)$ as 91858, hence both give $AB=\\sqrt{91858}$.",
"robustness_analysis": "Generality note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{91858}$.) |
math-000959 | Coordinate Geometry: Distance Formula | 1 | Where appropriate, name the theorem you use: Let $A(-67,-3)$ and $B(86,-32)$ 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": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(153,-29)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{24250}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(153,-29)$ as 24250, hence both give $AB=\\sqrt{24250}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{24250}$.) |
math-000960 | Coordinate Geometry: Distance Formula | 1 | Give reasoning, not just computation: Let $A(31,47)$ and $B(-128,-102)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the... | [
{
"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 $\\overrightarrow{AB}=(-159,-149)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{47482}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-159,-149)$ as 47482, hence both give $AB=\\sqrt{47482}$.",
"robustness_analysis": "Generality 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... | 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-000961 | Geometry: Pythagorean Theorem in Coordinates | 1 | Answer with a short justification: Let $A(147,111)$ and $B(-176,-178)$ 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 ... | [
{
"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 $\\overrightarrow{AB}=(-323,-289)$.",
"Step ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{187850}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-323,-289)$ as 187850, hence both give $AB=\\sqrt{187850}$.",
"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... | 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 $\boxed{\sqrt{187850}$.) |
math-000962 | Coordinate Geometry: Distance Formula | 1 | Carefully track domains: Let $A(-141,165)$ and $B(-117,9)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean ... | [
{
"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 $\\overrightarrow{AB}=(24,-156)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{24912}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(24,-156)$ as 24912, hence both give $AB=\\sqrt{24912}$.",
"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... | 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 $\boxed{\sqrt{24912}$.) |
math-000963 | Geometry: Pythagorean Theorem in Coordinates | 1 | Be explicit about assumptions: Let $A(196,-79)$ and $B(101,-146)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pytha... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=101-(196)=-95$ and $\\Delta y=-146-(-79)=-67$.",
"Step 2: A translation sen... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{13514}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-95,-67)$ as 13514, hence both give $AB=\\sqrt{13514}$.",
"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 $\boxed{\sqrt{13514}$.) |
math-000964 | Coordinate Geometry: Distance Formula | 1 | Answer using clear logical steps: Let $A(-175,-20)$ and $B(143,-18)$ 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 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 $\\overrightarrow{AB}=(318,2)$.",
"Step 2: C... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{101128}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(318,2)$ as 101128, hence both give $AB=\\sqrt{101128}$.",
"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 $\boxed{\sqrt{101128}$.) |
math-000965 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve and then verify: Let $A(-72,-101)$ and $B(46,79)$ 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": "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: $\\Delta x=46-(-72)=118$ and $\\Delta y=79-(-101)=180$.",
"Step 2: A translation sends... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{46324}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(118,180)$ as 46324, hence both give $AB=\\sqrt{46324}$.",
"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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{46324}$.) |
math-000966 | Geometry: Pythagorean Theorem in Coordinates | 1 | Find the exact value: Let $A(-59,-156)$ and $B(60,-108)$ 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 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: $\\Delta x=60-(-59)=119$ and $\\Delta y=-108-(-156)=48$.",
"Step 2: A translation send... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{16465}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(119,48)$ as 16465, hence both give $AB=\\sqrt{16465}$.",
"robustness_analysis": "Generality no... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000967 | Analytic Geometry: Translation Invariance | 1 | Solve with verification: Let $A(170,-171)$ and $B(-136,100)$ 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 $\\overrightarrow{AB}=(-306,271)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{167077}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-306,271)$ as 167077, hence both give $AB=\\sqrt{167077}$.",
"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... | 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-000968 | Analytic Geometry: Translation Invariance | 1 | Solve and include a self-check: Let $A(188,73)$ and $B(-70,-85)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-70-(188)=-258$ and $\\Delta y=-85-(73)=-158$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{91528}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-258,-158)$ as 91528, hence both give $AB=\\sqrt{91528}$.",
"robustness_analysis": "Robustness note: The vector/dot-product me... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{91528}$.) |
math-000969 | Coordinate Geometry: Distance Formula | 1 | Explain each transformation: Let $A(-80,184)$ and $B(104,-170)$ 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 $\\overrightarrow{AB}=(184,-354)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{159172}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(184,-354)$ as 159172, hence both give $AB=\\sqrt{159172}$.",
"robustness_analysis": "If the problem were perturbed: The... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{159172}$.) |
math-000970 | Analytic Geometry: Translation Invariance | 1 | Use two approaches if possible: Let $A(-132,65)$ and $B(-86,39)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-86-(-132)=46$ and $\\Delta y=39-(65)=-26$.",
"Step 2: A translation sends ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{2792}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(46,-26)$ as 2792, hence both give $AB=\\sqrt{2792}$.",
"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... | 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 $\boxed{\sqrt{2792}$.) |
math-000971 | Coordinate Geometry: Distance Formula | 1 | Warm-up: Let $A(-148,-63)$ and $B(13,12)$ 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 $\\overrightarrow{AB}=(161,75)$.",
"Step 2: ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{31546}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(161,75)$ as 31546, hence both give $AB=\\sqrt{31546}$.",
"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 $\boxed{\sqrt{31546}$.) |
math-000972 | Coordinate Geometry: Distance Formula | 1 | Find the exact value: Let $A(64,28)$ and $B(-81,61)$ 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 theore... | [
{
"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 $\\overrightarrow{AB}=(-145,33)$.",
"Step 2:... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{22114}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-145,33)$ as 22114, hence both give $AB=\\sqrt{22114}$.",
"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... | 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 $\boxed{\sqrt{22114}$.) |
math-000973 | Coordinate Geometry: Distance Formula | 1 | Determine the requested value: Let $A(130,84)$ and $B(-97,161)$ 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 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: $\\Delta x=-97-(130)=-227$ and $\\Delta y=161-(84)=77$.",
"Step 2: A translation sends... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{57458}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-227,77)$ as 57458, hence both give $AB=\\sqrt{57458}$.",
"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-000974 | Coordinate Geometry: Distance Formula | 1 | Solve and then verify: Let $A(-64,-65)$ and $B(174,-138)$ 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 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 $\\overrightarrow{AB}=(238,-73)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{61973}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(238,-73)$ as 61973, hence both give $AB=\\sqrt{61973}$.",
"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 $\boxed{\sqrt{61973}$.) |
math-000975 | Analytic Geometry: Translation Invariance | 1 | Find the exact value: Let $A(-64,77)$ and $B(-7,28)$ 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 theore... | [
{
"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: $\\Delta x=-7-(-64)=57$ and $\\Delta y=28-(77)=-49$.",
"Step 2: A translation sends $A... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{5650}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(57,-49)$ as 5650, hence both give $AB=\\sqrt{5650}$.",
"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... | 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 $\boxed{\sqrt{5650}$.) |
math-000976 | Vector Geometry: Norms and Dot Products | 1 | Start by stating any domain restrictions: Let $A(-198,-187)$ and $B(-2,191)$ 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: $\\Delta x=-2-(-198)=196$ and $\\Delta y=191-(-187)=378$.",
"Step 2: A translation sen... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{181300}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(196,378)$ as 181300, hence both give $AB=\\sqrt{181300}$.",
"robustness_analysis": "Robustness note: The vector/dot-pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{181300}$.) |
math-000977 | Vector Geometry: Norms and Dot Products | 1 | Derive the result step-by-step: Let $A(132,22)$ and $B(-142,95)$ 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 Pythag... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(-274,73)$.",
"Step 2:... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{80405}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-274,73)$ as 80405, hence both give $AB=\\sqrt{80405}$.",
"robustness_analysis": "Generality n... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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-000978 | Coordinate Geometry: Distance Formula | 1 | Start by stating any domain restrictions: Let $A(195,-84)$ and $B(141,160)$ 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... | [
{
"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 $\\overrightarrow{AB}=(-54,244)$.",
"Step 2:... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{62452}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-54,244)$ as 62452, hence both give $AB=\\sqrt{62452}$.",
"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 $\boxed{\sqrt{62452}$.) |
math-000979 | Coordinate Geometry: Distance Formula | 1 | Problem: Let $A(-88,-96)$ and $B(145,-199)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem 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 $\\overrightarrow{AB}=(233,-103)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{64898}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(233,-103)$ as 64898, hence both give $AB=\\sqrt{64898}$.",
"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... | 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 $\boxed{\sqrt{64898}$.) |
math-000980 | Coordinate Geometry: Distance Formula | 1 | Work this out carefully: Let $A(36,-82)$ and $B(-166,117)$ 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 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 $\\overrightarrow{AB}=(-202,199)$.",
"Step 2... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{80405}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-202,199)$ as 80405, hence both give $AB=\\sqrt{80405}$.",
"robustness_analysis": "If the problem were perturbed: The ve... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{80405}$.) |
math-000981 | Geometry: Pythagorean Theorem in Coordinates | 1 | Provide both a computational and a conceptual explanation: Let $A(141,-187)$ and $B(138,-107)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should r... | [
{
"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: $\\Delta x=138-(141)=-3$ and $\\Delta y=-107-(-187)=80$.",
"Step 2: A translation send... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{6409}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-3,80)$ as 6409, hence both give $AB=\\sqrt{6409}$.",
"robustness_analysis": "If the problem were perturbed: The vector/dot-pro... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{6409}$.) |
math-000982 | Coordinate Geometry: Distance Formula | 1 | Start by stating any domain restrictions: Let $A(46,107)$ and $B(146,-151)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=146-(46)=100$ and $\\Delta y=-151-(107)=-258$.",
"Step 2: A translation sen... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{76564}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(100,-258)$ as 76564, hence both give $AB=\\sqrt{76564}$.",
"robustness_analysis": "If the problem were perturbed: The vector/d... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Remember: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000983 | Coordinate Geometry: Distance Formula | 1 | Determine the requested value: Let $A(131,200)$ and $B(-115,-117)$ 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 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: $\\Delta x=-115-(131)=-246$ and $\\Delta y=-117-(200)=-317$.",
"Step 2: A translation ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{161005}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-246,-317)$ as 161005, hence both give $AB=\\sqrt{161005}$.",
"robustness_analysis": "Robustn... | [
{
"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 $\boxed{\sqrt{161005}$.) |
math-000984 | Coordinate Geometry: Distance Formula | 1 | Task: Let $A(-115,0)$ and $B(176,-19)$ 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 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: $\\Delta x=176-(-115)=291$ and $\\Delta y=-19-(0)=-19$.",
"Step 2: A translation sends... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{85042}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(291,-19)$ as 85042, hence both give $AB=\\sqrt{85042}$.",
"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... | 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-000985 | Geometry: Pythagorean Theorem in Coordinates | 1 | Proceed methodically: Let $A(-175,-155)$ and $B(-16,-138)$ 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 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: $\\Delta x=-16-(-175)=159$ and $\\Delta y=-138-(-155)=17$.",
"Step 2: A translation se... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{25570}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(159,17)$ as 25570, hence both give $AB=\\sqrt{25570}$.",
"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... | 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 $\boxed{\sqrt{25570}$.) |
math-000986 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve (and briefly cross-validate): Let $A(162,-108)$ and $B(81,115)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or 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: $\\Delta x=81-(162)=-81$ and $\\Delta y=115-(-108)=223$.",
"Step 2: A translation send... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{56290}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-81,223)$ as 56290, hence both give $AB=\\sqrt{56290}$.",
"robustness_analysis": "Sensitivity analysis: The vector/dot-p... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000987 | Coordinate Geometry: Distance Formula | 1 | Exercise: Let $A(-119,123)$ and $B(-122,-70)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem 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: $\\Delta x=-122-(-119)=-3$ and $\\Delta y=-70-(123)=-193$.",
"Step 2: A translation se... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{37258}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-3,-193)$ as 37258, hence both give $AB=\\sqrt{37258}$.",
"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 $\boxed{\sqrt{37258}$.) |
math-000988 | Analytic Geometry: Translation Invariance | 1 | State any required conditions first: Let $A(-187,-87)$ and $B(-145,2)$ 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 ... | [
{
"method_name": "Pythagorean Theorem",
"approach": "Translate the segment into a right triangle with legs $|\\Delta x|$ and $|\\Delta y|$; then apply Pythagoras.",
"steps": [
"Step 1: Compute differences: $\\Delta x=-145-(-187)=42$ and $\\Delta y=2-(-87)=89$.",
"Step 2: A translation sends ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{9685}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(42,89)$ as 9685, hence both give $AB=\\sqrt{9685}$.",
"robustness_analysis": "Generality note: 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 $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{9685}$.) |
math-000989 | Analytic Geometry: Translation Invariance | 1 | Solve (and briefly cross-validate): Let $A(144,-161)$ and $B(-166,-163)$ 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: $\\Delta x=-166-(144)=-310$ and $\\Delta y=-163-(-161)=-2$.",
"Step 2: A translation s... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{96104}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-310,-2)$ as 96104, hence both give $AB=\\sqrt{96104}$.",
"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... | 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-000990 | Geometry: Pythagorean Theorem in Coordinates | 1 | Solve with verification: Let $A(110,-84)$ and $B(-45,10)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=-45-(110)=-155$ and $\\Delta y=10-(-84)=94$.",
"Step 2: A translation sends... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{32861}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-155,94)$ as 32861, hence both give $AB=\\sqrt{32861}$.",
"robustness_analysis": "Generality note: The vector/dot-produc... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. |
math-000991 | Analytic Geometry: Translation Invariance | 1 | Find the exact value: Let $A(54,-74)$ and $B(130,2)$ 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 theore... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(76,76)$.",
"Step 2: C... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{11552}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(76,76)$ as 11552, hence both give $AB=\\sqrt{11552}$.",
"robustness_analysis": "Generality not... | [
{
"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 $\boxed{\sqrt{11552}$.) |
math-000992 | Analytic Geometry: Translation Invariance | 1 | Keep the final answer in boxed form: Let $A(-140,148)$ and $B(-173,-31)$ 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 $\\overrightarrow{AB}=(-33,-179)$.",
"Step 2... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{33130}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-33,-179)$ as 33130, hence both give $AB=\\sqrt{33130}$.",
"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 $\boxed{\sqrt{33130}$.) |
math-000993 | Analytic Geometry: Translation Invariance | 1 | Give a fully justified solution: Let $A(71,37)$ and $B(-45,3)$ 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 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: $\\Delta x=-45-(71)=-116$ and $\\Delta y=3-(37)=-34$.",
"Step 2: A translation sends $... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{14612}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-116,-34)$ as 14612, hence both give $AB=\\sqrt{14612}$.",
"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 $\boxed{\sqrt{14612}$.) |
math-000994 | Vector Geometry: Norms and Dot Products | 1 | Explain each transformation: Let $A(-46,83)$ and $B(-111,-29)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-65,-112)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{16769}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-65,-112)$ as 16769, hence both give $AB=\\sqrt{16769}$.",
"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... | 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-000995 | Coordinate Geometry: Distance Formula | 1 | Challenge: Let $A(107,119)$ and $B(153,158)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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: $\\Delta x=153-(107)=46$ and $\\Delta y=158-(119)=39$.",
"Step 2: A translation sends ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{3637}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(46,39)$ as 3637, hence both give $AB=\\sqrt{3637}$.",
"robustness_analysis": "Generality note: The vector/dot-product method ge... | [
{
"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 $\boxed{\sqrt{3637}$.) |
math-000996 | Coordinate Geometry: Distance Formula | 1 | Prompt: Let $A(-6,-141)$ and $B(172,127)$ 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 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 $\\overrightarrow{AB}=(178,268)$.",
"Step 2:... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\sqrt{103508}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(178,268)$ as 103508, hence both give $AB=\\sqrt{103508}$.",
"robustness_analysis": "If the problem were perturbed: The vector... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{103508}$.) |
math-000997 | Vector Geometry: Norms and Dot Products | 1 | Solve and justify each step: Let $A(-92,178)$ and $B(-182,26)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-90,-152)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{31204}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-90,-152)$ as 31204, hence both give $AB=\\sqrt{31204}$.",
"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... | 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 $\boxed{\sqrt{31204}$.) |
math-000998 | Geometry: Pythagorean Theorem in Coordinates | 1 | Prompt: Let $A(101,34)$ and $B(161,82)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the Pythagorean theorem explicitly.... | [
{
"method_name": "Vector Norm / Dot Product",
"approach": "Use $\\overrightarrow{AB}=(\\Delta x,\\Delta y)$ and compute $\\|\\overrightarrow{AB}\\|^2=\\overrightarrow{AB}\\cdot\\overrightarrow{AB}$.",
"steps": [
"Step 1: The displacement vector is $\\overrightarrow{AB}=(60,48)$.",
"Step 2: C... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\sqrt{5904}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(60,48)$ as 5904, hence both give $AB=\\sqrt{5904}$.",
"robustness_analysis": "Sensitivity analysis: The ... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | 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 $\boxed{\sqrt{5904}$.) |
math-000999 | Vector Geometry: Norms and Dot Products | 1 | Explain what is being counted/optimized: Let $A(-125,87)$ and $B(-175,-129)$ 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": "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 $\\overrightarrow{AB}=(-50,-216)$.",
"Step 2... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\sqrt{49156}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-50,-216)$ as 49156, hence both give $AB=\\sqrt{49156}$.",
"robustness_analysis": "Sensitivity... | [
{
"error_description": "Used $\\sqrt{\\Delta x+\\Delta y}$ instead of $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}$.",
"why_plausible": "Both formulas involve a square root and coordinate differences.",
"why_wrong": "Distance must be nonnegative and symmetric; without squares the expression can be negative or v... | Core principle: Distance depends only on the displacement vector between points, which explains translation invariance and yields $AB=\sqrt{(\Delta x)^2+(\Delta y)^2}$. (Here the result is $\boxed{\sqrt{49156}$.) |
math-001000 | Vector Geometry: Norms and Dot Products | 1 | Compute the requested quantity: Let $A(117,160)$ and $B(-125,-58)$ be points in the Euclidean plane.
(a) Compute the distance $AB$.
(b) Compute the squared distance $AB^2$.
(c) Explain why the formula is invariant under translations of the coordinate system.
Your explanation in (c) should reference vectors or the 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 $\\overrightarrow{AB}=(-242,-218)$.",
"Step ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\sqrt{106088}$.\nBoth methods compute the squared norm of the displacement $(\\Delta x,\\Delta y)=(-242,-218)$ as 106088, hence both give $AB=\\sqrt{106088}$.",
"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... | 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 $\boxed{\sqrt{106088}$.) |
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