COMC COMC Geometry and Combinatorics 1 — Questions and Answers
Question 1: What is the area of a triangle with vertices at (0, 0), (4, 0), and (0, 3)?
- 6 (Correct answer)
- 12
- 7
- 10
Correct answer: 6
The triangle has base 4 and height 3 along the coordinate axes, so its area is (1/2)(4)(3) = 6.
Question 2: A chord is 4 units from the center of a circle with radius 5. What is the length of the chord?
- 6 (Correct answer)
- 3
- 8
- 4
Correct answer: 6
The perpendicular bisects the chord; half-chord = √(5² − 4²) = √9 = 3, so the chord length is 6.
Question 3: Two angles of a triangle measure 45° and 75°. What is the measure of the third angle?
- 60° (Correct answer)
- 50°
- 70°
- 80°
Correct answer: 60°
The interior angles of a triangle sum to 180°, so the third angle is 180° − 45° − 75° = 60°.
Question 4: In a right triangle, one leg is 7 and the hypotenuse is 25. What is the length of the other leg?
- 24 (Correct answer)
- 18
- 20
- 16
Correct answer: 24
By the Pythagorean theorem, the other leg is √(25² − 7²) = √(625 − 49) = √576 = 24.
Question 5: Two similar triangles have corresponding sides in the ratio 3:5. What is the ratio of their areas?
- 9:25 (Correct answer)
- 3:5
- 6:10
- 27:125
Correct answer: 9:25
Areas of similar figures scale as the square of the linear ratio: (3/5)² = 9/25.
Question 6: What is the area of a regular hexagon with side length 2?
- 6√3 (Correct answer)
- 4√3
- 8√3
- 12
Correct answer: 6√3
A regular hexagon with side s has area (3√3/2)s²; with s = 2 the area is (3√3/2)(4) = 6√3.
COMC COMC Geometry and Combinatorics 1
What is the area of a triangle with vertices at (0, 0), (4, 0), and (0, 3)?