An altitude of an acute triangle divides its base into two adjacent segments of lengths 5 … — GRE General Test

An altitude of an acute triangle divides its base into two adjacent segments of lengths $5$ and $9$ . If the area of the triangle is $84$ square units, what is the perimeter of the triangle?

A
$28$
B
$36$
$42$ Answer
D
$48$
E
$52$

Answer

The perimeter of the triangle is

42

The base of the triangle is the sum of the two adjacent segments,

5 + 9 = 14

. Setting up the area formula gives

84 = \frac{1}{2} \times 14 \times h

, which yields an altitude height of

h = 12

. The altitude creates two right triangles: one with legs

5

and

12

(hypotenuse

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

) and another with legs

9

and

12

(hypotenuse

\sqrt{9^2 + 12^2} = 15

). Adding all three side lengths (

14 + 13 + 15

) yields

42

Step-by-Step Solution

Find the total base length and calculate the height (altitude) of the triangle.

Base

= 5 + 9 = 14

. Height

h = \frac{2 \times \text{Area}}{\text{base}} = \frac{2 \times 84}{14} = 12

The area of a triangle is given by

A = \frac{1}{2} b h

Calculate the lengths of the two non-base sides using the Pythagorean theorem on the two right triangles formed by the altitude.

Left side

= \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = 13

. Right side

= \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = 15

An altitude drawn to the base creates two right triangles sharing the altitude as a common leg.

Sum all three boundary sides to find the perimeter.

Perimeter

= 14 + 13 + 15 = 42

Perimeter is the total length around the outside of the triangle.

Key Concept

Area and perimeter of triangles split by an altitude using the Pythagorean theorem.

Estimated Time:1m 30s

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