PSAT Math Multiple-Choice Question 80: Answer and Explanation

Question: 80

In the triangle above, AB = BC. Which of the following accurately expresses the perimeter of the triangle?

• A. 10 + 10 sin 55°
• B. 10 + 10 cos 35°
• C. 10 +
• D. 25 tan 35°

Correct Answer: C

Explanation:

C

The question asks for an expression for the perimeter of a triangle. Use the geometry basic approach. Start by labeling the figure with the given information. Because AB = BC, ABC is an isosceles triangle, which has two sides that are equal. In an isosceles triangle, angles that are opposite equal sides must be equal. Therefore, ∠A and ∠C have the same measure, so ∠A is also 35°. Indicate that in the diagram and draw a line from point B to AC to create two right triangles:

AB and BC are equal hypotenuses of the right triangles. Since the two triangles are congruent, the base of 10 is split equally so that each right triangle has a base of 5. The side adjacent to the 35° angle is known, so use cosine. SOHCAHTOA says that cosine θ = . Therefore, cos 35° = . Solve to get AB = . Add AB + BC + AC = for the perimeter, but that's not an answer. Notice that two answer choices use a 55° angle, which is the angle at the top of each right triangle (180 - 90 - 35). Another way of saying cos 35° is sin 55° because the side that is adjacent to the 35° angle is opposite the 55° angle. Replace cos 35° with sin 55°. The correct answer is (C).