Drill 3 · Math · Geometry
ACT Math: Geometry (Drill 3) is a Math practice drill covering Geometry. It contains 5 original questions developed by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.
This ACT Geometry drill covers applying the 30-60-90 and 45-45-90 special right triangle ratios, computing trapezoid area, finding the lateral surface area of a cylinder, using slopes to identify parallel and perpendicular lines, and calculating arc length and sector area.
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?
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?
Explanation: Area of a trapezoid = ½(b₁ + b₂)(h) = ½(8 + 14)(6) = ½(22)(6) = 66. Choice B (132) drops the ½: (8 + 14)(6) = 132. Choice D (84) uses only the longer base, as if the figure were a rectangle: 14 × 6 = 84. Choice C (168) extends that error to twice the height: 14 × 12 = 168. Only Choice A applies the formula correctly.
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?
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?
Explanation: First find the slope of line ℓ: mℓ = (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?
Explanation: Sector area = (central angle / 360°) × πr² = (150/360) × π(12²) = (5/12)(144π) = 60π. Choice A (30π) comes from using 75° instead of 150°: (75/360)(144π) = 30π. Choice C (72π) comes from using 180° instead of 150°: (180/360)(144π) = 72π. Choice D (144π) is the area of the full circle, π(12²), with the sector fraction left out.