PSAT Math Multiple-Choice Question 79: Answer and Explanation

Question: 79

Note: Figure not drawn to scale.

In the figure above, if AB = 5, AC = 13, and DE = 24, what is the value of BD?

• A. 12
• B. 10
• C. 8
• D. 5

Correct Answer: D

Explanation:

D

The question asks for a length of a line segment in a figure containing triangles. Use the geometry basic approach. Start by labeling the figure with the given information. When given two or more triangles and information about the lengths of the sides, look for similar triangles. Both triangles share ∠DAE and each has a right angle. Since all triangles have 180°, the third angles in each triangle must also be equal. The two triangles must have the same set of angles, but they aren't the same size; they are similar triangles, so the sides of one triangle are proportional to those of the other. BD = AD - AB, and AB is given; find AD to get the answer. In the small right triangle, two sides are given, so use the Pythagorean Theorem to find the third: a² + b² = c². Plug in AB for a and AC for c so that 5² + b² = 13². Then solve for b, which will be BC, the length of the other side of the right triangle. First, simplify: 25 + b² = 169. Then subtract 25 from both sides and take the square root: . Therefore, b = 12. Another way to solve for b would be to recognize the Pythagorean triple: 5-12-13. Since BC = 12, set up a proportion with corresponding sides: or . Cross-multiply: (AD) (12) = 120. Divide both sides by 12 to get AD = 10. Now solve BD = AD - AB = 10 - 5 = 5. The correct answer is (D).