COMC COMC Geometry and Combinatorics 1

Question 1

What is the area of a triangle with vertices at (0, 0), (4, 0), and (0, 3)?

Accepted Answer

6

Answer

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?

Accepted Answer

6

Answer

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?

Accepted Answer

60°

Answer

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?

Accepted Answer

24

Answer

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?

Accepted Answer

9:25

Answer

Areas of similar figures scale as the square of the linear ratio: (3/5)² = 9/25.

COMC - Canadian Open Mathematics Challenge Practice Test