An isosceles triangle has a base of length 10 inches and a perimeter of 36 inches. What is… — GRE General Test

An isosceles triangle has a base of length $10$ inches and a perimeter of $36$ inches. What is the area of the triangle, in square inches?

60

square inches

Subtracting the base length of

10

inches from the total perimeter of

36

inches leaves

26

inches for the remaining two congruent sides, giving

13

inches each. The perpendicular height divides the isosceles triangle into two right triangles with base

5

inches and hypotenuse

13

inches. By the Pythagorean theorem, the height is

\sqrt{13^2 - 5^2} = 12

inches. Substituting base

10

and height

12

into the area formula

\frac{1}{2} \times \text{base} \times \text{height}

gives

\frac{1}{2} \times 10 \times 12 = 60

square inches.

Determine the length of the two congruent sides of the isosceles triangle.

Each side has a length of

13

inches.

Since the perimeter is

36

inches and the base is

10

inches, the sum of the two equal sides is

36 - 10 = 26

inches. Dividing by

2

yields

13

inches per side.

Calculate the perpendicular height of the triangle using the Pythagorean theorem.

The height of the triangle is

12

inches.

Dropping an altitude from the top vertex to the base bisects the base into two segments of

5

inches. This creates two right triangles with a base of

5

inches and hypotenuse of

13

inches. Thus,

h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

inches.

Calculate the area of the isosceles triangle.

The area is

60

square inches.

Using the triangle area formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, we compute

\frac{1}{2} \times 10 \times 12 = 60

Properties of Isosceles Triangles and Area Formula

Question