Geometry Similarity & Proportionality 3 — Questions and Answers
Question 1: Two similar solids have a scale factor of 2. What is the ratio of their volumes?
- 8:1 (Correct answer)
- 2:1
- 4:1
- 6:1
Correct answer: 8:1
Volume ratio is the cube of the scale factor: 2³ = 8.
Question 2: In a triangle, the angle bisector divides the opposite side into segments proportional to the:
- Adjacent sides (Correct answer)
- Opposite angles
- Other two angle bisectors
- Median lengths
Correct answer: Adjacent sides
The Angle Bisector Theorem states the ratio equals the ratio of the two adjacent sides.
Question 3: If DE is parallel to BC in triangle ABC with AD = 4, DB = 6, and AE = 6, what is EC?
- 9 (Correct answer)
- 4
- 6
- 10
Correct answer: 9
By proportionality 4/6 = 6/EC, so EC = 9.
Question 4: Two similar pentagons have areas 16 cm² and 81 cm². What is the ratio of their sides?
- 4:9 (Correct answer)
- 16:81
- 2:3
- 8:27
Correct answer: 4:9
Take the square root of the area ratio: √16:√81 = 4:9.
Question 5: Which of the following always produces similar figures?
- A dilation (Correct answer)
- A shear
- A stretch in one direction
- A random distortion
Correct answer: A dilation
A dilation preserves shape and angles, producing a similar figure.
Question 6: The altitude to the hypotenuse of a right triangle creates two smaller triangles that are:
- Similar to the original and each other (Correct answer)
- Congruent to each other
- Equilateral
- Isosceles
Correct answer: Similar to the original and each other
The altitude-on-hypotenuse rule makes all three triangles similar.
Question 7: A map uses a scale of 1 inch = 25 miles. Two cities are 3.5 inches apart. What is the actual distance?
- 87.5 miles (Correct answer)
- 71.4 miles
- 28.5 miles
- 75 miles
Correct answer: 87.5 miles
Multiply 3.5 × 25 = 87.5 miles.
Geometry Similarity & Proportionality 3
Two similar solids have a scale factor of 2.
What is the ratio of their volumes?