In \triangle ABC, the length of side AB is 13 units, the length of side BC is 21 units, an… — GRE General Test

In $\triangle ABC$ , the length of side $AB$ is $13$ units, the length of side $BC$ is $21$ units, and the area of $\triangle ABC$ is $126$ square units. If $\angle ABC$ is an acute angle, what is the length of side $AC$ ?

A
15
B
18
20Answer
D
22
E
25

Answer

The area of triangle

ABC

\frac{1}{2} \times 21 \times h = 126

, which yields an altitude

AH = 12

perpendicular to side

BC

. In right triangle

ABH

, the base segment

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

. Because angle

ABC

is acute,

H

falls between

B

and

C

, making

HC = 21 - 5 = 16

. Finally, in right triangle

AHC

AC = \sqrt{12^2 + 16^2} = \sqrt{400} = 20

. Therefore, 20 is the correct answer.

Step-by-Step Solution

Calculate the altitude

h

from vertex

A

to base

BC

h = 12

units.

The area formula for a triangle is

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

. Substituting

BC = 21

and

\text{Area} = 126

yields

126 = \frac{1}{2} \times 21 \times h \implies h = 12

Find the length of segment

BH

where

H

is the foot of the altitude on

BC

BH = 5

units.

In right triangle

\triangle ABH

AB = 13

and

AH = 12

. By the Pythagorean theorem,

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

Determine the length of segment

HC

HC = 16

units.

Since

\angle ABC

is an acute angle, point

H

lies on segment

BC

. Therefore,

HC = BC - BH = 21 - 5 = 16

Calculate the length of side

AC

AC = 20

units.

In right triangle

\triangle AHC

AH = 12

and

HC = 16

. By the Pythagorean theorem,

AC = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20

Key Concept

Triangles: Altitude, Area, and Pythagorean Theorem

Estimated Time:1m 30s

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