PSAT Math Multiple-Choice Question 699: Answer and Explanation

Question: 699

In two similar isosceles triangles, triangle A has two sides each of length 5 and one side of length 7. Triangle B has exactly one side of length 28. What is the perimeter of triangle B ?

• A. 17
• B. 20
• C. 38
• D. 68

Correct Answer: D

Explanation:

(D) Similar triangles have similar side lengths, meaning that the side lengths vary in fixed proportions. We know that triangle B has exactly one side length of 28. Since exactly one side of triangle A has length 7, this is 28 ÷ 7 = 4 times the side length of the unique side in triangle A . Thus, the two shorter sides in triangle B will also be 4 times the side length of the shorter sides in triangle A . Since the two other sides of triangle A have length 5, triangle B has two sides of length 4(5) = 20 and one side of length of 28.

This could have also been determined using a proportion:

Cross multiplication yields:

(28)(5) = 7x

140 = 7x

Dividing both sides by 7 tells us that 20 = x.

Here we need to be careful. Notice that choice (B) is 20, so you may be tempted to pick choice (B). However, the question is asking us for the perimeter of the triangle rather than for the unknown side length.

The perimeter is 20 + 20 + 28 = 68, choice (D).

An alternative approach would have been to recognize that the perimeters of similar triangles will vary in the same proportion as the side lengths. We know that triangle B has sides 4 times longer than those of triangle A , so triangle B will also have a perimeter 4 times that of triangle A.

Triangle A has a perimeter of 5 + 5 + 7 = 17.

Triangle B therefore has a perimeter of 4(17) = 68.