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Similarity Concepts

Grade 8 · Geometry · Worksheet 1

  1. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(6, 3), and C(2, 7). This triangle is first reflected across the y-axis, then dilated by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final image of vertex C after both transformations? Answer: ______________
  2. Triangle ABC has vertices A(1, 1), B(3, 1), C(1, 5). Triangle DEF has vertices D(3, 3), E(9, 3), F(3, 15). Describe a sequence of transformations that shows triangle ABC is similar to triangle DEF. Answer: ______________
  3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The garden is dilated by a scale factor of 2.5 with the origin as the center of dilation. What are the coordinates of the vertices of the dilated garden? Answer: ______________
  4. A quadrilateral has vertices at A(7, 8), B(15, 8), C(15, 12), and D(7, 12). It is first rotated 90 degrees clockwise about the origin, then dilated by a scale factor of 3 with the origin as the center of dilation. What are the coordinates of the final image of vertex B? Answer: ______________
  5. Triangle ABC has vertices A(8, 12), B(14, 12), and C(8, 20). Triangle DEF has vertices D(4, 6), E(7, 6), and F(4, 10). Describe a sequence of transformations that maps triangle ABC onto triangle DEF, including a dilation and a rigid motion. Answer: ______________
  6. Triangle ABC has vertices A(2, 2), B(7, 2), and C(2, 7). Triangle DEF is the image of triangle ABC after a dilation centered at the origin with a scale factor of 3, followed by a translation 4 units left and 1 unit down. What are the coordinates of vertex F? Answer: ______________
  7. Noah is creating a digital art piece that starts with a triangle with vertices at (0, 0), (9, 0), and (0, 7). He first dilates the triangle by a scale factor of 4 with the center at the origin. Then he translates the dilated triangle 3 units to the left and 5 units down. What are the coordinates of the final triangle's vertices? Answer: ______________
  8. (4.2 × 10^5) ÷ (7 × 10^2) = ? Answer: ______________
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Answer Key & Explanations

Similarity Concepts · Grade 8 · Worksheet 1

  1. A triangle is drawn on a coordinate plane with vertices at A(2, 3), B(6, 3), and C(2, 7). This triangle is first reflected across the y-axis, then dilated by a scale factor of 2 with the origin as the center of dilation. What are the coordinates of the final image of vertex C after both transformations? Answer: (-4, 14) Solution: Start with the original coordinates of vertex C: (2, 7) Apply reflection across the y-axis. Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same.
    Full step-by-step solution

    Step 1: Start with the original coordinates of vertex C: (2, 7) Step 2: Apply reflection across the y-axis. Reflection across the y-axis changes the sign of the x-coordinate while keeping the y-coordinate the same. So (2, 7) becomes (-2, 7). Step 3: Apply dilation with scale factor 2 and center at the origin. Dilation multiplies both coordinates by the scale factor. So (-2, 7) becomes (-2 × 2, 7 × 2) = (-4, 14). Step 4: The final coordinates after both transformations are (-4, 14).

  2. Triangle ABC has vertices A(1, 1), B(3, 1), C(1, 5). Triangle DEF has vertices D(3, 3), E(9, 3), F(3, 15). Describe a sequence of transformations that shows triangle ABC is similar to triangle DEF. Answer: Dilate by a scale factor of 3 with center at the origin, then translate by (2, 2). Solution: Find the side lengths of triangle ABC. AB = 3 - 1 = 2, AC = 5 - 1 = 4, BC = sqrt((3-1)^2 + (1-5)^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5). Step 2: Find the side lengths of triangle DEF.
    Full step-by-step solution

    Step 1: Find the side lengths of triangle ABC. AB = 3 - 1 = 2, AC = 5 - 1 = 4, BC = sqrt((3-1)^2 + (1-5)^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5). Step 2: Find the side lengths of triangle DEF. DE = 9 - 3 = 6, DF = 15 - 3 = 12, EF = sqrt((9-3)^2 + (3-15)^2) = sqrt(36 + 144) = sqrt(180) = 6*sqrt(5). Step 3: Compare side lengths: DE/AB = 6/2 = 3, DF/AC = 12/4 = 3, EF/BC = (6*sqrt(5))/(2*sqrt(5)) = 3. The scale factor is 3. Step 4: Dilate triangle ABC by a scale factor of 3 with center at the origin: A'(3, 3), B'(9, 3), C'(3, 15). Step 5: Compare A' to D: (3, 3) matches D(3, 3), so no translation needed. Actually, A' = D, B' = E, C' = F. So the sequence is: dilate by a scale factor of 3 with center at the origin. The answer is: Dilate by a scale factor of 3 with center at the origin.

  3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The garden is dilated by a scale factor of 2.5 with the origin as the center of dilation. What are the coordinates of the vertices of the dilated garden? Answer: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) Solution: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The center of dilation is the origin (0, 0). The scale factor is 2.5.
    Full step-by-step solution

    Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangle with vertices: A = (2, 1) B = (8, 1) C = (8, 5) D = (2, 5) The center of dilation is the origin (0, 0). The scale factor is 2.5. Dilation rule with center at origin: (x, y) → (2.5 * x, 2.5 * y) --- **Step 2: Apply dilation to each vertex** For A = (2, 1): New x = 2.5 * 2 = 5 New y = 2.5 * 1 = 2.5 A' = (5, 2.5) For B = (8, 1): New x = 2.5 * 8 = 20 New y = 2.5 * 1 = 2.5 B' = (20, 2.5) For C = (8, 5): New x = 2.5 * 8 = 20 New y = 2.5 * 5 = 12.5 C' = (20, 12.5) For D = (2, 5): New x = 2.5 * 2 = 5 New y = 2.5 * 5 = 12.5 D' = (5, 12.5) --- **Step 3: Write the final answer in order** The dilated rectangle's vertices are: (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5) --- **Step 4: Check** Original rectangle: width = 8 - 2 = 6, height = 5 - 1 = 4. After dilation: width = 20 - 5 = 15, height = 12.5 - 2.5 = 10. Scale factor check: 15/6 = 2.5, 10/4 = 2.5. Correct. --- **Final Answer:** (5, 2.5), (20, 2.5), (20, 12.5), (5, 12.5)

  4. A quadrilateral has vertices at A(7, 8), B(15, 8), C(15, 12), and D(7, 12). It is first rotated 90 degrees clockwise about the origin, then dilated by a scale factor of 3 with the origin as the center of dilation. What are the coordinates of the final image of vertex B? Answer: (24, -45) Solution: Original vertex B is at (15, 8). Rotate 90 degrees clockwise about the origin. Rule: (x, y) → (y, -x).
    Full step-by-step solution

    Step 1: Original vertex B is at (15, 8). Step 2: Rotate 90 degrees clockwise about the origin. Rule: (x, y) → (y, -x). So (15, 8) becomes (8, -15). Step 3: Dilate by scale factor 3 with center at origin. Rule: (x, y) → (3x, 3y). So (8, -15) becomes (3 × 8, 3 × -15) = (24, -45). The final coordinates of vertex B are (24, -45).

  5. Triangle ABC has vertices A(8, 12), B(14, 12), and C(8, 20). Triangle DEF has vertices D(4, 6), E(7, 6), and F(4, 10). Describe a sequence of transformations that maps triangle ABC onto triangle DEF, including a dilation and a rigid motion. Answer: Dilate triangle ABC by a scale factor of 1/2 centered at A(8, 12), then translate 4 units left and 6 units down. Solution: Find the side lengths of triangle ABC. AB = 14 - 8 = 6, AC = 20 - 12 = 8, BC = sqrt((14-8)^2 + (12-20)^2) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. Find the side lengths of triangle DEF.
    Full step-by-step solution

    Step 1: Find the side lengths of triangle ABC. AB = 14 - 8 = 6, AC = 20 - 12 = 8, BC = sqrt((14-8)^2 + (12-20)^2) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 2: Find the side lengths of triangle DEF. DE = 7 - 4 = 3, DF = 10 - 6 = 4, EF = sqrt((7-4)^2 + (6-10)^2) = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = sqrt(25) = 5. Step 3: The ratio of corresponding sides is 3/6 = 4/8 = 5/10 = 1/2, so the scale factor is 1/2. Step 4: Choose a center of dilation. Since vertex A(8, 12) corresponds to D(4, 6), dilating with center A by scale factor 1/2 maps A to itself (since A is the center). Then B(14, 12) maps to (8 + 0.5*(14-8), 12 + 0.5*(12-12)) = (8 + 3, 12) = (11, 12). C(8, 20) maps to (8 + 0.5*(8-8), 12 + 0.5*(20-12)) = (8, 12 + 4) = (8, 16). Step 5: The dilated triangle has vertices A(8, 12), B'(11, 12), C'(8, 16). To map this to D(4, 6), E(7, 6), F(4, 10), we need a translation: subtract 4 from x and 6 from y. So translation vector is (-4, -6). Step 6: The sequence is: dilate by scale factor 1/2 with center A(8, 12), then translate by (-4, -6). The answer is: Dilate triangle ABC by a scale factor of 1/2 centered at A(8, 12), then translate 4 units left and 6 units down.

  6. Triangle ABC has vertices A(2, 2), B(7, 2), and C(2, 7). Triangle DEF is the image of triangle ABC after a dilation centered at the origin with a scale factor of 3, followed by a translation 4 units left and 1 unit down. What are the coordinates of vertex F? Answer: (-2, 20) Solution: Identify the original coordinates of vertex C: (2, 7). Apply dilation with scale factor 3 centered at the origin: multiply each coordinate by 3.
    Full step-by-step solution

    Step 1: Identify the original coordinates of vertex C: (2, 7). Step 2: Apply dilation with scale factor 3 centered at the origin: multiply each coordinate by 3. (2 × 3, 7 × 3) = (6, 21) Step 3: Apply translation 4 units left and 1 unit down: subtract 4 from the x-coordinate and subtract 1 from the y-coordinate. (6 - 4, 21 - 1) = (2, 20) Step 4: The coordinates of vertex F are (2, 20). The answer is (2, 20).

  7. Noah is creating a digital art piece that starts with a triangle with vertices at (0, 0), (9, 0), and (0, 7). He first dilates the triangle by a scale factor of 4 with the center at the origin. Then he translates the dilated triangle 3 units to the left and 5 units down. What are the coordinates of the final triangle's vertices? Answer: (-3, -5), (33, -5), (-3, 23) Solution: Original vertices: A(0, 0), B(9, 0), C(0, 7). Dilation by scale factor 4 centered at origin: Multiply each coordinate by 4.
    Full step-by-step solution

    Step 1: Original vertices: A(0, 0), B(9, 0), C(0, 7). Step 2: Dilation by scale factor 4 centered at origin: Multiply each coordinate by 4. A' = (0*4, 0*4) = (0, 0) B' = (9*4, 0*4) = (36, 0) C' = (0*4, 7*4) = (0, 28) Step 3: Translation 3 units left and 5 units down: Subtract 3 from x, subtract 5 from y. A'' = (0 - 3, 0 - 5) = (-3, -5) B'' = (36 - 3, 0 - 5) = (33, -5) C'' = (0 - 3, 28 - 5) = (-3, 23) Step 4: Final vertices: (-3, -5), (33, -5), (-3, 23).

  8. (4.2 × 10^5) ÷ (7 × 10^2) = ? Answer: 600 Solution: Separate the problem into two parts: (4.2 ÷ 7) and (10^5 ÷ 10^2). Calculate 4.2 ÷ 7 = 0.6. Calculate 10^5 ÷ 10^2 = 10^(5-2) = 10^3.
    Full step-by-step solution

    Step 1: Separate the problem into two parts: (4.2 ÷ 7) and (10^5 ÷ 10^2). Step 2: Calculate 4.2 ÷ 7 = 0.6. Step 3: Calculate 10^5 ÷ 10^2 = 10^(5-2) = 10^3. Step 4: Multiply the results: 0.6 × 10^3 = 0.6 × 1000 = 600. The answer is 600.