Similarity Concepts

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Answer Key & Explanations

Similarity Concepts · Grade 8 · Worksheet 2

1. Liam is designing a triangular logo for his robotics team. He starts with a triangle that has vertices at (0,0), (4,0), and (0,3). He then applies a transformation that multiplies all the coordinates by 2.5. After this, he reflects the new triangle across the y-axis. What are the coordinates of the final triangle's vertices? Answer: (-10,0), (0,0), (0,7.5) Solution: A = (0, 0) B = (4, 0) C = (0, 3) Multiply all coordinates by 2.5 This is a dilation (scaling) with factor 2.5.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Original triangle vertices** The original triangle has vertices: A = (0, 0) B = (4, 0) C = (0, 3) --- **Step 2: Multiply all coordinates by 2.5** This is a dilation (scaling) with factor 2.5. Multiply each coordinate of each vertex by 2.5: A' = (0 × 2.5, 0 × 2.5) = (0, 0) B' = (4 × 2.5, 0 × 2.5) = (10, 0) C' = (0 × 2.5, 3 × 2.5) = (0, 7.5) So after scaling: A' = (0, 0) B' = (10, 0) C' = (0, 7.5) --- **Step 3: Reflect across the y-axis** Reflecting across the y-axis changes the sign of the x-coordinate, while the y-coordinate stays the same. Rule: (x, y) → (-x, y) A'' = (-0, 0) = (0, 0) B'' = (-10, 0) C'' = (0, 7.5) (since x=0, reflection keeps it at 0) --- **Step 4: Final vertices** The final triangle's vertices are: (0, 0), (-10, 0), (0, 7.5) We can list them in any order, but the problem's correct answer is given as: (-10, 0), (0, 0), (0, 7.5) --- **Final Answer:** (-10, 0), (0, 0), (0, 7.5)

2. Sophia is designing a digital logo for a school event. She starts with a triangle on a coordinate grid with vertices at A(0, 0), B(14, 0), and C(0, 21). To create a similar but larger logo, she first dilates the triangle by a scale factor of 3/2 centered at the origin. Then, she translates the dilated triangle 4 units to the right and 3 units down. What are the final coordinates of vertex A? Answer: (4, -3) Solution: Start with the original vertex A at (0, 0). Multiply each coordinate by 3/2: (0 * 3/2, 0 * 3/2) = (0, 0). Add 4 to the x-coordinate and subtract 3 from the y-coordinate: (0 + 4, 0 - 3) = (4, -3).
   Full step-by-step solution

   Step 1: Start with the original vertex A at (0, 0). Step 2: Apply the dilation with scale factor 3/2 centered at the origin. Multiply each coordinate by 3/2: (0 * 3/2, 0 * 3/2) = (0, 0). Step 3: Apply the translation 4 units to the right and 3 units down. Add 4 to the x-coordinate and subtract 3 from the y-coordinate: (0 + 4, 0 - 3) = (4, -3). Step 4: The final coordinates of vertex A are (4, -3).

3. Triangle ABC has vertices A(3, 5), B(7, 5), and C(3, 9). Triangle DEF has vertices D(9, 15), E(21, 15), and F(9, 27). Describe a sequence of transformations that shows triangle ABC is similar to triangle DEF. Answer: Dilate triangle ABC by a scale factor of 3 centered at the origin, then translate 0 units (or no translation needed). Solution: Identify corresponding vertices. A(3,5) corresponds to D(9,15), B(7,5) to E(21,15), C(3,9) to F(9,27). Check for dilation.
   Full step-by-step solution

   Step 1: Identify corresponding vertices. A(3,5) corresponds to D(9,15), B(7,5) to E(21,15), C(3,9) to F(9,27). Step 2: Check for dilation. Compare coordinates of A and D: x-coordinate 3 to 9 is multiplied by 3, y-coordinate 5 to 15 is multiplied by 3. Check B to E: 7 to 21 (×3), 5 to 15 (×3). Check C to F: 3 to 9 (×3), 9 to 27 (×3). So a dilation by scale factor 3 centered at the origin maps ABC to a triangle with vertices (9,15), (21,15), (9,27). Step 3: After dilation, the triangle is exactly at the coordinates of triangle DEF. No translation, reflection, or rotation is needed. Step 4: Therefore, the sequence is: dilate triangle ABC by a scale factor of 3 with center at the origin. This shows similarity because dilation preserves shape and changes size, and no further rigid motion is required. The answer is: Dilate by a scale factor of 3 centered at the origin.

4. Triangle ABC has vertices at A(2, 3), B(6, 3), and C(4, 7). Triangle DEF is created by applying a dilation with center at the origin and a scale factor of 2.5 to triangle ABC. What is the length of side D'E' in triangle DEF? Answer: 10 Solution: Triangle ABC has vertices A(2, 3), B(6, 3), and C(4, 7). A dilation with center at the origin and scale factor 2.5 is applied to produce triangle DEF. We need the length of side D'E' in triangle DEF.
   Full step-by-step solution

   Step 1: Understand the problem Triangle ABC has vertices A(2, 3), B(6, 3), and C(4, 7). A dilation with center at the origin and scale factor 2.5 is applied to produce triangle DEF. We need the length of side D'E' in triangle DEF. Step 2: Identify corresponding sides D'E' corresponds to side AB in triangle ABC, because D corresponds to A, E corresponds to B, and F corresponds to C. Step 3: Find length AB A(2, 3) and B(6, 3) have the same y-coordinate, so AB is horizontal. Length AB = |6 - 2| = 4. Step 4: Apply dilation to length A dilation with scale factor k multiplies all lengths by k. Here k = 2.5. So D'E' = (scale factor) × AB = 2.5 × 4. Step 5: Calculate 2.5 × 4 = 10. Step 6: Conclusion The length of side D'E' in triangle DEF is 10. Final answer: 10

5. Mere is designing a triangular pattern for a traditional weaving project. She starts with a triangle that has vertices at (0, 0), (0, 15), and (20, 0) on a coordinate grid. She wants to create a larger, similar triangle by applying a dilation with a scale factor of 2.5 centered at the origin, and then translating the new triangle 10 units to the left. What are the coordinates of the final image of the vertex that was originally at (20, 0)? Answer: (40, 0) Solution: Identify the original vertex: (20, 0). Multiply each coordinate by 2.5: (20 * 2.5, 0 * 2.5) = (50, 0). Subtract 10 from the x-coordinate: (50 - 10, 0) = (40, 0).
   Full step-by-step solution

   Step 1: Identify the original vertex: (20, 0). Step 2: Apply the dilation with scale factor 2.5 centered at the origin. Multiply each coordinate by 2.5: (20 * 2.5, 0 * 2.5) = (50, 0). Step 3: Apply the translation 10 units to the left. Subtract 10 from the x-coordinate: (50 - 10, 0) = (40, 0). Step 4: The final coordinates of the vertex are (40, 0).

6. Triangle ABC has vertices A(8, 2), B(12, 2), C(10, 8). Triangle DEF has vertices D(4, 1), E(6, 1), F(5, 4). Describe the sequence of transformations (dilation and rigid motions) that maps triangle ABC onto triangle DEF. Answer: Dilate by a factor of 1/2 with center at the origin, then translate 0 units horizontally and -1 unit vertically (or simply dilate by 1/2 about the origin). Solution: Find the side lengths of triangle ABC. AB = 12 - 8 = 4, BC = sqrt((10-12)^2 + (8-2)^2) = sqrt(4 + 36) = sqrt(40) = 2*sqrt(10), AC = sqrt((10-8)^2 + (8-2)^2) = sqrt(4 + 36) = 2*sqrt(10).
   Full step-by-step solution

   Step 1: Find the side lengths of triangle ABC. AB = 12 - 8 = 4, BC = sqrt((10-12)^2 + (8-2)^2) = sqrt(4 + 36) = sqrt(40) = 2*sqrt(10), AC = sqrt((10-8)^2 + (8-2)^2) = sqrt(4 + 36) = 2*sqrt(10). So triangle ABC is isosceles with base AB = 4. Step 2: Find the side lengths of triangle DEF. DE = 6 - 4 = 2, DF = sqrt((5-4)^2 + (4-1)^2) = sqrt(1 + 9) = sqrt(10), EF = sqrt((5-6)^2 + (4-1)^2) = sqrt(1 + 9) = sqrt(10). So triangle DEF is isosceles with base DE = 2. Step 3: The ratio of corresponding sides is DE/AB = 2/4 = 1/2, and DF/AC = sqrt(10)/(2*sqrt(10)) = 1/2. So the scale factor from ABC to DEF is 1/2. Step 4: Apply a dilation with center at the origin and scale factor 1/2 to triangle ABC. A(8,2) -> A'(4,1), B(12,2) -> B'(6,1), C(10,8) -> C'(5,4). Step 5: Compare A'(4,1), B'(6,1), C'(5,4) with D(4,1), E(6,1), F(5,4). They match exactly. So no translation is needed. The sequence is: dilate triangle ABC by a factor of 1/2 with center at the origin.

7. Liam is designing a triangular logo for his school's robotics team. The original triangle has side lengths of 6 cm, 8 cm, and 10 cm. He creates a scaled version where the shortest side is 9 cm. What are the lengths of the other two sides in the scaled logo? Answer: 12 cm and 15 cm Solution: Identify the scale factor. The original triangle has sides 6 cm, 8 cm, and 10 cm. The shortest side is 6 cm.
   Full step-by-step solution

   Step 1: Identify the scale factor. The original triangle has sides 6 cm, 8 cm, and 10 cm. The shortest side is 6 cm. The scaled version has the shortest side as 9 cm. Step 2: Calculate the scale factor. Scale factor = (new shortest side) / (original shortest side) Scale factor = 9 / 6 = 3/2 = 1.5 Step 3: Apply the scale factor to the other sides. Original second side = 8 cm New second side = 8 × (3/2) = 24/2 = 12 cm Original third side = 10 cm New third side = 10 × (3/2) = 30/2 = 15 cm Step 4: State the final side lengths. The scaled logo has sides 9 cm, 12 cm, and 15 cm. ANSWER: 12 cm and 15 cm