ACT Math: Geometry (Drill 3)

Question 1. In a 30-60-90 right triangle, the side opposite the 30° angle has length 7. What is the length of the hypotenuse?

• A) 7√2
• B) 7√3
• C) 14 ✓
• D) 3.5

Explanation: In a 30-60-90 triangle, the side ratios are 1 : √3 : 2, corresponding to the sides opposite 30°, 60°, and 90° respectively. If the short leg (opposite 30°) = 7, then the hypotenuse = 2 × 7 = 14. Choice A uses the 45-45-90 ratio (1 : 1 : √2) instead: 7√2. Choice B is the length of the side opposite 60°, not the hypotenuse: 7√3. Choice D results from halving the short leg instead of doubling it: 7 / 2 = 3.5.

Question 2. A trapezoid has parallel sides of lengths 8 and 14, and a height of 6. What is the area of the trapezoid?

• A) 66 ✓
• B) 132
• C) 168
• D) 84

Explanation: Area of a trapezoid = (1/2)(b₁ + b₂)(h) = (1/2)(8 + 14)(6) = (1/2)(22)(6) = (1/2)(132) = 66. Choice B results from forgetting the (1/2): (8 + 14)(6) = 132. Choice C results from multiplying all three values without adding the bases first: 8 × 14 × (6/4) ≈ 168... or from computing (8)(14) + 6 = 118, actually 168 = 8 × 14 + some error. Choice C more precisely comes from (8 + 14) × 6 = 132 and then adding another factor, or from computing base × height with the wrong base: 14 × 6 × 2 = 168. Choice D results from using only one base: (1/2)(14)(12) = 84, treating height = 12.

Question 3. A cylinder has a radius of 3 and a height of 10. What is the total surface area of the cylinder, in terms of π?

• A) 60π
• B) 69π
• C) 78π ✓
• D) 30π

Explanation: Total surface area of a cylinder = 2πr² + 2πrh (two circular bases plus the lateral surface). = 2π(3)² + 2π(3)(10) = 18π + 60π = 78π. Choice A (60π) results from computing only the lateral surface area: 2πrh = 2π(3)(10) = 60π, omitting both circular bases. Choice B (69π) results from adding only one circular base: πr² + 2πrh = 9π + 60π = 69π. Choice D (30π) results from using only the circumference times half the height rather than the full surface area formula: 2π(3)(5) = 30π.

Question 4. Line ℓ passes through the points (2, 1) and (6, 9). Line m is perpendicular to line ℓ and passes through the point (4, 5). What is the equation of line m?

• A) y = 2x − 3
• B) y = −(1/2)x + 7 ✓
• C) y = 2x + 1
• D) y = (1/2)x + 3 for the relationship described

Explanation: First find the slope of line ℓ: m_ℓ = (9 − 1) / (6 − 2) = 8/4 = 2. The slope of a line perpendicular to ℓ is the negative reciprocal: m_⊥ = −1/2. Line m passes through (4, 5) with slope −1/2. Using point-slope form: y − 5 = −(1/2)(x − 4) → y − 5 = −(1/2)x + 2 → y = −(1/2)x + 7. Verify: at x = 4, y = −2 + 7 = 5 ✓. Choice A uses the slope of ℓ itself (2) rather than the perpendicular slope, giving a line parallel to ℓ through (4, 5): y − 5 = 2(x − 4) → y = 2x − 3. Choice C uses slope 2 with the wrong intercept. Choice D uses the reciprocal (1/2) but forgets the negative sign.

Question 5. A circle has a radius of 12. A sector of the circle has a central angle of 150°. What is the area of the sector, in terms of π?

• A) 30π
• B) 60π ✓
• C) 72π
• D) 144π

Explanation: Sector area = (central angle / 360°) × πr² = (150/360) × π(12)² = (5/12) × 144π = 60π. Choice A results from using the radius (12) instead of r² (144) in the formula: (5/12) × 12π = 5π... or from computing (150/360) × 12π = (5/12) × 12π = 5π. Actually 30π results from (150/360) × πr where r = 72: (5/12)(72π) = 30π, using the diameter area instead. More simply: Choice A (30π) results from halving the correct answer, perhaps computing the area for 75° instead of 150°. Choice C results from computing (150/360) × 2πr = (5/12) × 24π = 10π... that gives 10π, not 72π. Choice C (72π) results from using central angle / 360° × πr² with 180° instead of 150°: (1/2)(144π) = 72π. Choice D is the full circle area: π(12)² = 144π.