Question

In triangle ABC, angle B = 90°, cos(c) = 15/17, and AB = 16 units. Based on this information, m angle A = angle C = and AC =​

a) (m∠ A = 45°, , m∠ C = 45°, , AC = 17 , units)
b) (m∠ A = 60°, , m∠ C = 30°, , AC = 17 , units)
c) (m∠ A = 30°, , m∠ C = 60°, , AC = 15 , units)
d) (m∠ A = 45°, , m∠ C = 45°, , AC = 15 , units)

Answer 1

Given the cosine of angle C and length AB in a right triangle, we determine that angle A is 30°, angle C is 60°, and the length of AC is 17 units.

Step-by-step explanation:

In triangle ABC, we are given that angle B = 90° and cos(C) = 15/17, with AB = 16 units. Since triangle ABC is a right triangle, by definition, cos(C) is the adjacent side (AB) over the hypotenuse (AC). Therefore, AC must be equal to 17 units because 15/17 is a Pythagorean triple (15, 8, 17), representing the sides of a right triangle. With one angle being 90° and the cosine of another angle given, the triangle is determined. Since cos(30°) = √3/2, cos(C) = 15/17 does not match this value, indicating that C is not 30°. Instead, we can infer that A and C are complementary angles (adding up to 90°), and since cos(C) corresponds to an acute angle in a 3-4-5 multiplied by 4 (12-16-20) Pythagorean triple, we can conclude that angle A must be 30° and angle C must be 60°.

Therefore, the correct answer is: m∠ A = 30°, m∠ C = 60°, and AC = 17 units, which is option c).