Similarity Concepts

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Answer Key & Explanations

Similarity Concepts · Grade 8 · Worksheet 1

1. A rectangle has vertices at A(2, 1), B(8, 1), C(8, 4), and D(2, 4). A similar rectangle is created by applying a dilation with center at the origin and a scale factor of 3, followed by a translation 2 units up. What is the length of the longer side in the transformed rectangle? Answer: 18 Solution: Find the original side lengths of rectangle ABCD. From A(2,1) to B(8,1): horizontal distance = 8 - 2 = 6 From A(2,1) to D(2,4): vertical distance = 4 - 1 = 3 The longer side is 6 units.
   Full step-by-step solution

   Step 1: Find the original side lengths of rectangle ABCD. From A(2,1) to B(8,1): horizontal distance = 8 - 2 = 6 From A(2,1) to D(2,4): vertical distance = 4 - 1 = 3 The longer side is 6 units. Step 2: Apply the dilation with scale factor 3. Dilation multiplies all lengths by the scale factor. Longer side after dilation = 6 × 3 = 18 units Step 3: Apply the translation 2 units up. Translation does not change side lengths, only position. Step 4: The length of the longer side in the transformed rectangle is 18 units. The answer is 18.

2. Maya is creating a scale drawing of her school's triangular garden. The actual garden has side lengths of 24 feet, 30 feet, and 36 feet. For her drawing, she uses a scale where 2 inches represents 6 feet. What will be the length of the longest side in Maya's scale drawing, in inches? Answer: 12 Solution: Identify the scale factor. The scale is 2 inches : 6 feet, which simplifies to 1 inch : 3 feet. Identify the longest side of the actual garden, which is 36 feet.
   Full step-by-step solution

   Step 1: Identify the scale factor. The scale is 2 inches : 6 feet, which simplifies to 1 inch : 3 feet. Step 2: Identify the longest side of the actual garden, which is 36 feet. Step 3: Set up a proportion using the scale factor: drawing length / actual length = 1 inch / 3 feet. Step 4: Let x be the drawing length of the longest side. Then x / 36 = 1 / 3. Step 5: Solve for x: x = 36 × (1/3) = 36/3 = 12. Step 6: The longest side in the scale drawing is 12 inches.

3. (6.3 × 10⁴) × (2.5 × 10⁻³) ÷ (1.5 × 10²) = ? Answer: 1.05 Solution: Multiply the coefficients: 6.3 × 2.5 = 15.75 Multiply the powers of 10: 10⁴ × 10⁻³ = 10^(4 + (-3)) = 10¹ Now we have (15.75 × 10¹) ÷ (1.5 × 10²) Divide the coefficients: 15.75 ÷ 1.5 = 10.5 Divide the powers of 10: 10¹ ÷ 10² = 10^(1 - 2) = 10⁻¹ Combine the results: 10.5 × 10⁻¹ = 1.05 The answer…
   Full step-by-step solution

   Step 1: Multiply the coefficients: 6.3 × 2.5 = 15.75 Step 2: Multiply the powers of 10: 10⁴ × 10⁻³ = 10^(4 + (-3)) = 10¹ Step 3: Now we have (15.75 × 10¹) ÷ (1.5 × 10²) Step 4: Divide the coefficients: 15.75 ÷ 1.5 = 10.5 Step 5: Divide the powers of 10: 10¹ ÷ 10² = 10^(1 - 2) = 10⁻¹ Step 6: Combine the results: 10.5 × 10⁻¹ = 1.05 The answer is 1.05.

4. Liam is designing a triangular logo for his school club. He starts with a triangle that has vertices at (0,0), (4,0), and (0,3). He then applies a transformation that doubles the size of the triangle and moves it 2 units to the right. What are the coordinates of the vertices of the transformed triangle? Answer: (2,0), (10,0), (2,6) Solution: A = (0, 0) B = (4, 0) C = (0, 3) 1. "doubles the size of the triangle" — this means a dilation (scaling) by factor 2. 2.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the original vertices** The original triangle has vertices: A = (0, 0) B = (4, 0) C = (0, 3) --- **Step 2: Understand the transformation** The problem says: 1. "doubles the size of the triangle" — this means a dilation (scaling) by factor 2. 2. "moves it 2 units to the right" — this means a translation to the right by 2 units. We must decide the order. Usually, scaling happens first, then translation, unless otherwise specified. --- **Step 3: Apply scaling by factor 2** Scaling with factor 2 multiplies both x and y coordinates by 2. A' = (0 × 2, 0 × 2) = (0, 0) B' = (4 × 2, 0 × 2) = (8, 0) C' = (0 × 2, 3 × 2) = (0, 6) After scaling: A' = (0, 0) B' = (8, 0) C' = (0, 6) --- **Step 4: Apply translation 2 units to the right** Translation 2 units right means: add 2 to x-coordinate, y-coordinate unchanged. A'' = (0 + 2, 0) = (2, 0) B'' = (8 + 2, 0) = (10, 0) C'' = (0 + 2, 6) = (2, 6) --- **Step 5: Final vertices** (2, 0), (10, 0), (2, 6) --- **Final answer:** (2,0), (10,0), (2,6)

5. Liam is designing a logo for his robotics team. He starts with a triangle that has vertices at (0,0), (4,0), and (2,3). He wants to create a similar triangle that is larger, with the side corresponding to the base from (0,0) to (4,0) now stretching from (0,0) to (10,0). What are the coordinates of the third vertex of the enlarged triangle? Answer: (5, 7.5) Solution: A = (0, 0) B = (4, 0) C = (2, 3) The base AB is from (0,0) to (4,0), length = 4. The new triangle is similar to the original. The base in the new triangle is from (0,0) to (10,0), length = 10.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have an original triangle with vertices: A = (0, 0) B = (4, 0) C = (2, 3) The base AB is from (0,0) to (4,0), length = 4. The new triangle is similar to the original. The base in the new triangle is from (0,0) to (10,0), length = 10. The base in the new triangle corresponds to AB in the original. --- **Step 2: Find the scale factor** Scale factor = (new base length) / (original base length) = 10 / 4 = 2.5 So all lengths in the new triangle are 2.5 times the lengths in the original. --- **Step 3: Locate the third vertex in the original triangle** Original third vertex C = (2, 3). We can think of C relative to A = (0,0): Vector AC = (2 - 0, 3 - 0) = (2, 3). --- **Step 4: Apply the scale factor to the vector AC** Since the triangles are similar with A fixed at (0,0) and B going from (4,0) to (10,0), the scaling is centered at A? Actually, not necessarily centered at A, but the correspondence is: A → A' = (0,0) B → B' = (10,0) C → C' = ? In similar figures, coordinates relative to A scale by the same factor. So new vector A'C' = scale factor × vector AC = 2.5 × (2, 3) = (5, 7.5) Thus C' = A' + (5, 7.5) = (0 + 5, 0 + 7.5) = (5, 7.5) --- **Step 5: Verify with the other vertex correspondence** Check with B: Original vector BC = (2 - 4, 3 - 0) = (-2, 3) New B' = (10, 0) New C' = B' + scale factor × vector BC = (10, 0) + 2.5 × (-2, 3) = (10, 0) + (-5, 7.5) = (5, 7.5) Same result. --- **Final Answer:** The third vertex of the enlarged triangle is (5, 7.5).

6. √(8² + 6²) = ? Answer: 10 Solution: Calculate 8 squared: 8² = 64 Calculate 6 squared: 6² = 36 Add the squares: 64 + 36 = 100 Take the square root: √100 = 10 The answer is 10.
   Full step-by-step solution

   Step 1: Calculate 8 squared: 8² = 64 Step 2: Calculate 6 squared: 6² = 36 Step 3: Add the squares: 64 + 36 = 100 Step 4: Take the square root: √100 = 10 The answer is 10.

7. (4.5 × 10⁶) ÷ (1.5 × 10²) = ? Answer: 30000 Solution: Write the division of the two numbers in scientific notation: (4.5 × 10⁶) ÷ (1.5 × 10²) Separate the problem into two parts: (4.5 ÷ 1.5) and (10⁶ ÷ 10²) Calculate 4.5 ÷ 1.5 = 3 Calculate 10⁶ ÷ 10² = 10^(6-2) = 10⁴ Combine the results: 3 × 10⁴ Convert 3 × 10⁴ to standard form: 3 × 10000 = 30000…
   Full step-by-step solution

   Step 1: Write the division of the two numbers in scientific notation: (4.5 × 10⁶) ÷ (1.5 × 10²) Step 2: Separate the problem into two parts: (4.5 ÷ 1.5) and (10⁶ ÷ 10²) Step 3: Calculate 4.5 ÷ 1.5 = 3 Step 4: Calculate 10⁶ ÷ 10² = 10^(6-2) = 10⁴ Step 5: Combine the results: 3 × 10⁴ Step 6: Convert 3 × 10⁴ to standard form: 3 × 10000 = 30000 The answer is 30000.

8. (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.