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Similarity Concepts

Grade 8 · Geometry · Worksheet 3

  1. Triangle ABC has vertices A(1, 1), B(5, 1), and C(1, 5). Triangle DEF has vertices D(3, 3), E(15, 3), and F(3, 15). Describe a sequence of transformations that shows triangle DEF is similar to triangle ABC. Answer: ______________
  2. (3.6 × 10⁴) × (2.5 × 10⁻³) ÷ (1.2 × 10²) = ? Answer: ______________
  3. Triangle ABC has vertices A(9, 3), B(12, 3), and C(9, 15). Triangle DEF has vertices D(3, 1), E(4, 1), and F(3, 5). Describe a sequence of transformations (dilation and rigid motions) that maps triangle DEF onto triangle ABC. Answer: ______________
  4. Triangle ABC has vertices A(1, 1), B(5, 1), C(3, 5). Triangle DEF has vertices D(3, 3), E(15, 3), F(9, 15). Describe a sequence of transformations (a dilation followed by a rigid motion) that maps triangle ABC onto triangle DEF. Answer: ______________
  5. Triangle ABC has vertices A(1, 1), B(1, 6), and C(6, 1). Triangle DEF has vertices D(2, 2), E(2, 12), and F(12, 2). Describe a sequence of transformations (dilation followed by a rigid motion) that maps triangle ABC onto triangle DEF. Answer: ______________
  6. (3 × 10⁸) ÷ (6 × 10⁴) = ? Answer: ______________
  7. Triangle ABC has vertices A(3, 1), B(7, 1), and C(3, 5). Triangle DEF has vertices D(9, 3), E(21, 3), and F(9, 15). Describe a sequence of transformations (a dilation followed by a rigid motion) that maps triangle ABC onto triangle DEF. Answer: ______________
  8. Hana is designing a logo on a coordinate grid. She starts with a triangle with vertices at A(2, 0), B(4, 0), and C(2, 6). To create a similar but larger logo, she first dilates the triangle by a scale factor of 2 with the origin as the center of dilation. Then, she reflects the dilated triangle across the x-axis. What are the coordinates of the final image of vertex C after both transformations? Answer: ______________
  9. (4.2 × 10³) × (3.5 × 10²) ÷ (2.1 × 10⁴) = ? Answer: ______________
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Answer Key & Explanations

Similarity Concepts · Grade 8 · Worksheet 3

  1. Triangle ABC has vertices A(1, 1), B(5, 1), and C(1, 5). Triangle DEF has vertices D(3, 3), E(15, 3), and F(3, 15). Describe a sequence of transformations that shows triangle DEF is similar to triangle ABC. Answer: Dilate triangle ABC 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 = 5 - 1 = 4, AC = 5 - 1 = 4, BC = sqrt((5-1)^2 + (5-1)^2) = sqrt(16+16) = sqrt(32) = 4*sqrt(2). So triangle ABC is an isosceles right triangle.
    Full step-by-step solution

    Step 1: Find the side lengths of triangle ABC. AB = 5 - 1 = 4, AC = 5 - 1 = 4, BC = sqrt((5-1)^2 + (5-1)^2) = sqrt(16+16) = sqrt(32) = 4*sqrt(2). So triangle ABC is an isosceles right triangle. Step 2: Find the side lengths of triangle DEF. DE = 15 - 3 = 12, DF = 15 - 3 = 12, EF = sqrt((15-3)^2 + (15-3)^2) = sqrt(144+144) = sqrt(288) = 12*sqrt(2). So triangle DEF is also an isosceles right triangle. Step 3: Compare corresponding sides: DE/AB = 12/4 = 3, DF/AC = 12/4 = 3, EF/BC = (12*sqrt(2))/(4*sqrt(2)) = 3. The scale factor is 3. Step 4: To map triangle ABC to triangle DEF, first dilate by a scale factor of 3 with center at the origin. A(1,1) becomes (3,3), B(5,1) becomes (15,3), C(1,5) becomes (3,15). This gives the exact coordinates of triangle DEF. Step 5: Since the dilated triangle already matches triangle DEF exactly, no translation is needed. The sequence is: dilate by a scale factor of 3 with center at the origin. The answer is: Dilate triangle ABC by a scale factor of 3 with center at the origin.

  2. (3.6 × 10⁴) × (2.5 × 10⁻³) ÷ (1.2 × 10²) = ? Answer: 0.75 Solution: Multiply the coefficients: 3.6 × 2.5 = 9.0 Multiply the powers of 10: 10⁴ × 10⁻³ = 10^(4 + (-3)) = 10¹ Now we have (9.0 × 10¹) ÷ (1.2 × 10²) Divide the coefficients: 9.0 ÷ 1.2 = 7.5 Divide the powers of 10: 10¹ ÷ 10² = 10^(1 - 2) = 10⁻¹ Combine the results: 7.5 × 10⁻¹ = 0.75 The answer is 0.75.
    Full step-by-step solution

    Step 1: Multiply the coefficients: 3.6 × 2.5 = 9.0 Step 2: Multiply the powers of 10: 10⁴ × 10⁻³ = 10^(4 + (-3)) = 10¹ Step 3: Now we have (9.0 × 10¹) ÷ (1.2 × 10²) Step 4: Divide the coefficients: 9.0 ÷ 1.2 = 7.5 Step 5: Divide the powers of 10: 10¹ ÷ 10² = 10^(1 - 2) = 10⁻¹ Step 6: Combine the results: 7.5 × 10⁻¹ = 0.75 The answer is 0.75.

  3. Triangle ABC has vertices A(9, 3), B(12, 3), and C(9, 15). Triangle DEF has vertices D(3, 1), E(4, 1), and F(3, 5). Describe a sequence of transformations (dilation and rigid motions) that maps triangle DEF onto triangle ABC. Answer: Dilate triangle DEF by a scale factor of 3 centered at the origin, then translate 6 units right and 0 units up (or simply dilate by 3 centered at the origin). Solution: Find the side lengths of triangle DEF. DE = |4 - 3| = 1, DF = |5 - 1| = 4, EF = sqrt((4-3)^2 + (1-5)^2) = sqrt(1 + 16) = sqrt(17). Step 2: Find the side lengths of triangle ABC.
    Full step-by-step solution

    Step 1: Find the side lengths of triangle DEF. DE = |4 - 3| = 1, DF = |5 - 1| = 4, EF = sqrt((4-3)^2 + (1-5)^2) = sqrt(1 + 16) = sqrt(17). Step 2: Find the side lengths of triangle ABC. AB = |12 - 9| = 3, AC = |15 - 3| = 12, BC = sqrt((12-9)^2 + (3-15)^2) = sqrt(9 + 144) = sqrt(153) = sqrt(9*17) = 3*sqrt(17). Step 3: Compare corresponding sides: AB/DE = 3/1 = 3, AC/DF = 12/4 = 3, BC/EF = 3*sqrt(17)/sqrt(17) = 3. So the scale factor is 3. Step 4: Dilate triangle DEF by a scale factor of 3 centered at the origin: D'(3*3, 3*1) = (9, 3), E'(3*4, 3*1) = (12, 3), F'(3*3, 3*5) = (9, 15). Step 5: The dilated triangle D'E'F' has vertices (9,3), (12,3), (9,15) which exactly match triangle ABC. No translation or rotation is needed. The sequence is: dilate by a scale factor of 3 centered at the origin.

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

    Step 1: Find the side lengths of triangle ABC. AB = 5 - 1 = 4, BC = sqrt((5-3)^2 + (1-5)^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5), AC = sqrt((3-1)^2 + (5-1)^2) = sqrt(4 + 16) = sqrt(20) = 2*sqrt(5). Step 2: Find the side lengths of triangle DEF. DE = 15 - 3 = 12, EF = sqrt((15-9)^2 + (3-15)^2) = sqrt(36 + 144) = sqrt(180) = 6*sqrt(5), DF = sqrt((9-3)^2 + (15-3)^2) = sqrt(36 + 144) = sqrt(180) = 6*sqrt(5). Step 3: The ratio of corresponding sides is 12/4 = 3, so the scale factor is 3. Step 4: Dilate triangle ABC by a factor of 3 with center at the origin: A'(3, 3), B'(15, 3), C'(9, 15). Step 5: Compare A' to D: D is at (3, 3), same as A'. B' to E: E is at (15, 3), same as B'. C' to F: F is at (9, 15), same as C'. No translation is needed; the dilation alone maps ABC onto DEF. The answer is: Dilate by a factor of 3 with center at the origin.

  5. Triangle ABC has vertices A(1, 1), B(1, 6), and C(6, 1). Triangle DEF has vertices D(2, 2), E(2, 12), and F(12, 2). Describe a sequence of transformations (dilation followed by a rigid motion) that maps triangle ABC onto triangle DEF. Answer: Dilate by a factor of 2 with center at the origin, then translate 0 units horizontally and 0 units vertically (or no translation needed). Solution: Identify the coordinates of triangle ABC: A(1,1), B(1,6), C(6,1). Triangle DEF: D(2,2), E(2,12), F(12,2). Compare corresponding vertices.
    Full step-by-step solution

    Step 1: Identify the coordinates of triangle ABC: A(1,1), B(1,6), C(6,1). Triangle DEF: D(2,2), E(2,12), F(12,2). Step 2: Compare corresponding vertices. Notice that D = (2,2) = (2×1, 2×1) = 2×A. Similarly, E = (2,12) = (2×1, 2×6) = 2×B, and F = (12,2) = (2×6, 2×1) = 2×C. Step 3: This shows that triangle DEF is exactly twice the size of triangle ABC, with the same orientation and centered at the origin. So a dilation by a factor of 2 with center at the origin maps ABC to a triangle with vertices (2,2), (2,12), (12,2), which is exactly triangle DEF. Step 4: Since the dilated triangle already matches DEF exactly, no additional rigid motion (translation, rotation, or reflection) is needed. The answer: Dilate by a factor of 2 with center at the origin.

  6. (3 × 10⁸) ÷ (6 × 10⁴) = ? Answer: 5000 Solution: We have: (3 × 10⁸) ÷ (6 × 10⁴) Break the expression into two parts — the numerical coefficients and the powers of ten. (3 × 10⁸) ÷ (6 × 10⁴) = (3 ÷ 6) × (10⁸ ÷ 10⁴) Simplify the numerical part.
    Full step-by-step solution

    Let's solve step by step. We have: (3 × 10⁸) ÷ (6 × 10⁴) Step 1: Break the expression into two parts — the numerical coefficients and the powers of ten. (3 × 10⁸) ÷ (6 × 10⁴) = (3 ÷ 6) × (10⁸ ÷ 10⁴) Step 2: Simplify the numerical part. 3 ÷ 6 = 3/6 = 1/2 = 0.5 Step 3: Simplify the powers of ten using the rule: 10⁸ ÷ 10⁴ = 10^(8 - 4) = 10⁴. So now we have: 0.5 × 10⁴ Step 4: Multiply 0.5 by 10⁴. 0.5 × 10⁴ = 0.5 × 10000 = 5000 Step 5: Final answer. 5000 That’s the result.

  7. Triangle ABC has vertices A(3, 1), B(7, 1), and C(3, 5). Triangle DEF has vertices D(9, 3), E(21, 3), and F(9, 15). Describe a sequence of transformations (a dilation followed by a rigid motion) that maps triangle ABC onto triangle DEF. Answer: Dilate by a scale factor of 3 centered at the origin, then translate by (0, 2). Solution: Find the side lengths of triangle ABC. AB = 7 - 3 = 4, AC = 5 - 1 = 4, BC = sqrt((7-3)^2 + (1-5)^2) = sqrt(16 + 16) = sqrt(32) = 4*sqrt(2). So triangle ABC is an isosceles right triangle.
    Full step-by-step solution

    Step 1: Find the side lengths of triangle ABC. AB = 7 - 3 = 4, AC = 5 - 1 = 4, BC = sqrt((7-3)^2 + (1-5)^2) = sqrt(16 + 16) = sqrt(32) = 4*sqrt(2). So triangle ABC is an isosceles right triangle. Step 2: Find the side lengths of triangle DEF. DE = 21 - 9 = 12, DF = 15 - 3 = 12, EF = sqrt((21-9)^2 + (3-15)^2) = sqrt(144 + 144) = sqrt(288) = 12*sqrt(2). The side lengths of DEF are 3 times those of ABC, so the scale factor is 3. Step 3: Dilate triangle ABC by a scale factor of 3 centered at the origin. A(3,1) becomes A'(9,3), B(7,1) becomes B'(21,3), C(3,5) becomes C'(9,15). Step 4: Compare A'B'C' to DEF. A'(9,3) matches D(9,3), B'(21,3) matches E(21,3), C'(9,15) matches F(9,15). So the dilated triangle is already exactly at the position of DEF. No rigid motion is needed. The sequence is: dilate by a scale factor of 3 centered at the origin.

  8. Hana is designing a logo on a coordinate grid. She starts with a triangle with vertices at A(2, 0), B(4, 0), and C(2, 6). To create a similar but larger logo, she first dilates the triangle by a scale factor of 2 with the origin as the center of dilation. Then, she reflects the dilated triangle across the x-axis. What are the coordinates of the final image of vertex C after both transformations? Answer: (4, -12) Solution: Start with the original coordinates of vertex C: (2, 6). Apply dilation with scale factor 2 and center at the origin. Dilation multiplies both coordinates by 2.
    Full step-by-step solution

    Step 1: Start with the original coordinates of vertex C: (2, 6). Step 2: Apply dilation with scale factor 2 and center at the origin. Dilation multiplies both coordinates by 2. So (2, 6) becomes (2 x 2, 6 x 2) = (4, 12). Step 3: Apply reflection across the x-axis. Reflection across the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate. So (4, 12) becomes (4, -12). Step 4: The final coordinates of vertex C after both transformations are (4, -12).

  9. (4.2 × 10³) × (3.5 × 10²) ÷ (2.1 × 10⁴) = ? Answer: 70 Solution: First, multiply the coefficients: 4.2 × 3.5 = 14.7 Add the exponents for the multiplication: 3 + 2 = 5 So we have 14.7 × 10^5 Now divide by (2.1 × 10^4): 14.7 ÷ 2.1 = 7 Subtract the exponents: 5 - 4 = 1 This gives us 7 × 10^1 = 70 The answer is 70.
    Full step-by-step solution

    Step 1: First, multiply the coefficients: 4.2 × 3.5 = 14.7 Step 2: Add the exponents for the multiplication: 3 + 2 = 5 Step 3: So we have 14.7 × 10^5 Step 4: Now divide by (2.1 × 10^4): 14.7 ÷ 2.1 = 7 Step 5: Subtract the exponents: 5 - 4 = 1 Step 6: This gives us 7 × 10^1 = 70 The answer is 70.