ACT Math: Geometry (Drill 2)

Question 1. A circle has a radius of 7. What is the ratio of its circumference to its area? (Express your answer in simplified form.)

• A) 2/7 ✓
• B) 2/49
• C) 7/2
• D) 1/7

Explanation: Circumference = 2πr = 2π(7) = 14π. Area = πr² = π(49) = 49π. Ratio = 14π / 49π = 14/49 = 2/7. The π cancels, making the ratio independent of π. Choice B (2/49) results from using just 2π as the circumference instead of 2πr: 2π / 49π = 2/49. Choice C (7/2) results from inverting the ratio, computing area/circumference instead of circumference/area: 49π / 14π = 7/2. Choice D (1/7) results from using πr instead of 2πr for the circumference, omitting the factor of 2: 7π / 49π = 1/7.

Question 2. Two parallel lines are cut by a transversal. One of the interior angles formed on the same side of the transversal measures 112°. What is the measure of the other interior angle on the same side of the transversal?

• A) 68° ✓
• B) 78°
• C) 112°
• D) 248°

Explanation: When two parallel lines are cut by a transversal, co-interior angles (also called consecutive interior angles or same-side interior angles) are supplementary; they add up to 180°. So the other angle = 180° − 112° = 68°. Choice B results from subtracting from 90° instead of 180°: treating the angles as complementary rather than supplementary. Choice C results from confusing co-interior angles with alternate interior angles, alternate interior angles are equal (112°), not supplementary. Choice D results from adding 112° + 136° = 248°, which has no geometric basis.

Question 3. A rectangular box has a length of 8 inches, a width of 5 inches, and a height of 4 inches. If the length is doubled and the height is halved, by what factor does the volume change?

• A) The volume is unchanged ✓
• B) The volume doubles
• C) The volume is halved
• D) The volume is multiplied by 4

Explanation: Original volume = 8 × 5 × 4 = 160 in³. New dimensions: length = 16, width = 5, height = 2. New volume = 16 × 5 × 2 = 160 in³. The volume is unchanged. This makes sense algebraically: multiplying one dimension by 2 and dividing another by 2 gives a net factor of 2 × (1/2) = 1, so the volume stays the same. Choice B results from focusing on the doubled length and ignoring the halved height. Choice C results from focusing on the halved height and ignoring the doubled length. Choice D results from doubling both the length and the height instead of halving the height.

Question 4. Triangle ABC is similar to Triangle DEF. In Triangle ABC, the sides have lengths AB = 6, BC = 10, and AC = 8. In Triangle DEF, DE = 9. What is the length of EF?

• A) 12
• B) 13
• C) 15 ✓
• D) 18

Explanation: In similar triangles, corresponding sides are proportional. AB corresponds to DE, so the scale factor from Triangle ABC to Triangle DEF is DE / AB = 9 / 6 = 3/2. BC corresponds to EF, so EF = BC × (3/2) = 10 × (3/2) = 15. Choice A (12) results from using AC (8) instead of BC (10) as the side corresponding to EF: 8 × (3/2) = 12. Choice B (13) results from adding the difference in corresponding sides to BC rather than applying the scale factor: 10 + (9 − 6) = 10 + 3 = 13. Choice D (18) results from multiplying DE by 2 rather than applying the scale factor to BC: 9 × 2 = 18.

Question 5. Which of the following is the equation of a circle with center (−3, 5) and radius 4?

• A) (x + 3)² + (y − 5)² = 16 ✓
• B) (x − 3)² + (y + 5)² = 16
• C) (x + 3)² + (y − 5)² = 4
• D) (x − 3)² + (y − 5)² = 16

Explanation: The standard form of a circle is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. With center (−3, 5) and r = 4: (x − (−3))² + (y − 5)² = 4², which simplifies to (x + 3)² + (y − 5)² = 16. Choice B results from using the center coordinates directly as written without applying the sign rule: center (−3, 5) written as (x − 3) and (y + 5). Choice C has the correct center but uses r = 4 instead of r² = 16 on the right side. Choice D uses (x − 3) instead of (x + 3), placing the center at (3, 5) instead of (−3, 5).