The right triangle ABC is shown. What is the value of tan A?

Explanation

Choice C is correct. In the triangle shown, the measure of angle B is 30° and angle C is a right angle, which means that it has a measure of 90°. Since the sum of the angles in a triangle is equal to 180°, the measure of angle A is equal to 180° - (30 + 90)o, or 60o. In a right triangle whose acute angles have measures 30° and 60°, the lengths of the legs can be represented by the expressions x, x √3, and 2x, where x is the length of the leg opposite the angle with measure 30°, x √3 is the length of the leg opposite the angle with measure 60°, and 2x is the length of the hypotenuse. In the triangle shown, the hypotenuse has a length of 54. It follows that 2x = 54, or x = 27. Therefore, the length of the leg opposite angle B is 27, and the length of the leg opposite angle A is 27 √3. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle A is 27 √3 and the length of the leg adjacent to angle A is 27. Therefore, the value of tan A is (27√3/27), or √3.