Congruence, Dilations and Similarity: Grade 8 Math Test Answer Key – PDF Download

Question 1: Move a point (2 points)

1. \((-3+6,\ 5-4)=\) \((3,1)\).
2. \((x,y)\to(x,-y)\) gives \((-3,-5)\).

Scoring: 1 point for each correct image. Give no credit if the signs are wrong.

Question 2: Rotate a point (2 points)

1. \((x,y)\to(-y,x)\): \((2,4)\).
2. \((x,y)\to(-x,-y)\): \((-4,2)\).

Scoring: 1 point for each correct image.

Question 3: Dilate two points (2 points)

1. \((\frac{6}{3},\ \frac{-9}{3})=\) \((2,-3)\).
2. \((4\cdot(-2),\ 4\cdot5)=\) \((-8,20)\).

Scoring: 1 point for each correct image.

Question 4: Reflect and compare (3 points)

1. Use \((x,y)\to(-x,y)\): \(A^{\prime}(-1,2)\), \(B^{\prime}(-5,2)\), \(C^{\prime}(-1,5)\).
2. Yes. A reflection is a rigid motion, so it keeps all side lengths and angle measures.

Scoring: part (a) 2 points, 1 point if two of three vertices are correct; part (b) 1 point for yes with a reason that names rigid motions or preserved lengths and angles.

Question 5: Dilation of a triangle (2 points)

1. \(5\cdot3=15\), \(12\cdot3=36\), \(13\cdot3=39\): \(15\text{ ft}\), \(36\text{ ft}\), \(39\text{ ft}\).
2. \(90^{\circ}\). A dilation does not change angle measures.

Scoring: 1 point for all three lengths (0 if only two are right); 1 point for \(90^{\circ}\) with the reason.

Question 6: Find a scale factor (2 points)

1. \(12: 3=4\) and \(-8: (-2)=4\): \(k=4\).
2. \((4\cdot(-5),\ 4\cdot6)=\) \(B^{\prime}(-20,24)\).

Scoring: 1 point for each part.

Question 7: Show similarity (3 points)

1. Side \(AB=2\) and \(DE=4\), so \(k=2\). Dilating gives \((2,2)\), \((6,2)\), \((2,8)\). Reflecting across the \(x\)-axis gives \((2,-2)\), \((6,-2)\), \((2,-8)\). Dilate by 2, then reflect across the \(x\)-axis.
2. Similar, not congruent. A dilation with \(k=2\) was needed, so the sizes differ.

Scoring: part (a) 1 point for the dilation with \(k=2\) and 1 point for the reflection across the \(x\)-axis; part (b) 1 point for similar and not congruent with a reason.

Question 8: Similar rectangles (2 points)

1. \(8: 6=\frac{4}{3}\) and \(12: 9=\frac{4}{3}\). Yes, the scale factor is \(\frac{4}{3}\).
2. \(10: 6=\frac{5}{3}\) but \(16: 9=\frac{16}{9}\). No, the ratios are different.

Scoring: 1 point for each part (correct answer with ratios shown).

Question 9: Two rigid motions (2 points)

1. After the translation: \((-1,-1)\), \((2,-1)\), \((2,3)\). After the reflection: \((1,-1)\), \((-2,-1)\), \((-2,3)\).
2. 5 units. Both moves are rigid motions, so lengths do not change.

Scoring: 1 point for the three final vertices (partial credit not given); 1 point for 5 with the reason.

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