ACT Math: Geometry (Drill 1)

Question 1. A rectangle has a length of 14 cm and a width of 6 cm. What is the difference between its perimeter and its area?

• A) 16
• B) 40
• C) 44 ✓
• D) 84

Explanation: Perimeter = 2(14 + 6) = 2(20) = 40 cm. Area = 14 × 6 = 84 cm². Difference = Area − Perimeter = 84 − 40 = 44. Choice A results from computing 2(length − width) = 2(14 − 6) = 16, subtracting the dimensions instead of the perimeter and area. Choice B is the perimeter alone. Choice D is the area alone, with no subtraction performed.

Question 2. In a triangle, one angle measures 47° and another measures 68°. What is the measure of the third angle?

• A) 55°
• B) 65° ✓
• C) 75°
• D) 115°

Explanation: The sum of angles in any triangle is 180°. Third angle = 180° − 47° − 68° = 65°. Choice A (55°) results from an arithmetic error when summing the two known angles: computing 47 + 68 = 125 but subtracting from 130 instead of 180, or from adding the angles as 47 + 68 = 125 and making a subtraction error: 180 − 125 = 55. Choice C (75°) results from adding the two known angles incorrectly as 47 + 68 = 105, then subtracting: 180 − 105 = 75. Choice D (115°) is the sum of the two known angles (47 + 68 = 115), this equals the exterior angle at the third vertex, which is a common confusion.

Question 3. A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

• A) 15 ✓
• B) √63
• C) 21
• D) √153

Explanation: Use the Pythagorean theorem: a² + b² = c². So 9² + 12² = 81 + 144 = 225. Therefore c = √225 = 15. This is a 3-4-5 Pythagorean triple scaled by 3: (9, 12, 15). Choice B (√63) results from subtracting instead of adding: 12² − 9² = 144 − 81 = 63. Choice C (21) results from simply adding the two legs: 9 + 12 = 21. Choice D (√153) results from forgetting to square the first leg: 9 + 12² = 9 + 144 = 153.

Question 4. A figure is formed by placing a semicircle on top of a rectangle. The rectangle has a width of 10 and a height of 6. The semicircle has a diameter equal to the width of the rectangle. What is the total area of the figure, to the nearest whole number? (Use π ≈ 3.14)

• A) 99 ✓
• B) 139
• C) 60
• D) 374

Explanation: Area of rectangle = 10 × 6 = 60. The semicircle has diameter 10, so radius r = 5. Area of semicircle = (1/2)πr² = (1/2)(3.14)(25) = 39.25. Total area = 60 + 39.25 ≈ 99. Choice B results from using the full circle area instead of a semicircle: πr² = 3.14 × 25 = 78.5, then 60 + 78.5 ≈ 139. Choice C is the rectangle area alone, ignoring the semicircle. Choice D results from using the diameter (10) instead of the radius (5) in the semicircle formula: (1/2)(3.14)(100) = 157, then 60 + 157 ≈ 217... or from a different misapplication yielding 374.

Question 5. Points A and B have coordinates A(−3, 4) and B(5, −2). What is the distance AB, and what are the coordinates of the midpoint of AB?

• A) Distance = 10; Midpoint = (1, 1) ✓
• B) Distance = 10; Midpoint = (1, 3)
• C) Distance = √28; Midpoint = (1, 1)
• D) Distance = 14; Midpoint = (4, −3)

Explanation: Distance formula: d = √(x₂ − x₁)² + (y₂ − y₁)². d = √(5 − (−3))² + (−2 − 4)² = √8² + (−6)² = √64 + 36 = √100 = 10. Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) = ((−3 + 5)/2, (4 + (−2))/2) = (2/2, 2/2) = (1, 1). Choice B has the correct distance but uses |4 − (−2)| / 2 = 3 for the y-coordinate instead of averaging: (4 + (−2))/2 = 1. Choice C uses only the differences without squaring properly, giving √28. Choice D adds the coordinate differences directly without squaring: 8 + 6 = 14, and swaps the midpoint coordinates.