Question

The altitude drawn to the base of an isosceles triangle is $8,$ and the perimeter is $32.$ The area of the triangle is:

$56$

$48$

$40$

$32$

$24$

Answer: B

Concepts:isosceles triangle Pythagorean Theorem triangle area

Difficulty rating: 1550

Small Hint:

Let each equal side be $a$ and half the base be $b$

Big Hint:

Use $a+b=16$ and $a^2-b^2=8^2$

Solution:

Let each equal side be $a$ and half the base be $b.$ The perimeter gives $a+b=16.$ The altitude bisects the base, so $a^2-b^2=8^2.$ Thus $(a-b)(a+b)=64,$ giving $a-b=4.$ Hence $b=6,$ so the base is $12.$ The area is $\frac12(12)(8)=48.$

Therefore, the correct answer is B.

1958 AMC 12 Problem 31

Answer: B

Difficulty rating: 1550

Small Hint:

Big Hint:

Solution:

Problem 31 in Other Years