An isosceles triangle has a perimeter of 40 centimeters, and... | SAT Lines, angles, and triangles Question, Answer Explained

Question bank / Lines, angles, and triangles

Math · Geometry and TrigonometryHard

An isosceles triangle has a perimeter of $40$ centimeters, and one of its sides has a length of $10$ centimeters. What are the lengths, in centimeters, of the other two sides?

$15$ and $15$

$10$ and $20$

Either $10$ and $20$ or $15$ and $15$

$12$ and $18$

Show answer and explanation

Answer: A

If $10$ is the length of each of the two equal sides, the third side is $40 - 20 = 20$ . But $10 + 10$ is not greater than $20$ , so these lengths can't form a triangle. So $10$ must be the length of the third side, and each equal side has length $\frac{40 - 10}{2} = 15$ . The lengths $15$ , $15$ , and $10$ do form a triangle.

Choice C seems to cover both cases, but sides of $10$ , $10$ , and $20$ break the triangle inequality, since $10 + 10$ is not greater than $20$ .

Why the other choices are wrong

B. Sides of 10, 10, and 20 can't form a triangle, because 10 + 10 is not greater than 20.
C. Only one case works. Sides of 10, 10, and 20 break the triangle inequality.
D. These lengths give a perimeter of 40, but the triangle wouldn't be isosceles.

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