In triangle ABC, the length of side AB is 15 and the length of side BC is 25. The area of … — GRE General Test

In triangle $ABC$ , the length of side $AB$ is $15$ and the length of side $BC$ is $25$ . The area of triangle $ABC$ is $150$ . If angle $ABC$ is obtuse, what is the length of side $AC$ ?

A
$20$
B
$5\sqrt{34}$
$10\sqrt{13}$ Answer
D
$34$
E
$40$

Answer

10\sqrt{13}

The correct answer

10\sqrt{13}

is obtained by drawing altitude

AH = 12

to line

BC

. In right triangle

ABH

, the base projection is

BH = \sqrt{15^2 - 12^2} = 9

. Because angle

ABC

is obtuse, point

H

lies outside segment

BC

, so

CH = 25 + 9 = 34

. Applying the Pythagorean theorem to right triangle

AHC

gives

AC = \sqrt{12^2 + 34^2} = \sqrt{1300} = 10\sqrt{13}

Step-by-Step Solution

Calculate the height (altitude)

h

perpendicular to line

BC

Since

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

, we have

150 = \frac{1}{2} \times 25 \times h

, which simplifies to

h = 12

The area formula for any triangle connects the base length and its corresponding perpendicular altitude.

Determine the location of the altitude foot

H

and calculate segment

BH

Draw altitude

AH

to the line containing

BC

. In right triangle

ABH

AB = 15

and

AH = 12

, so

BH = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9

The Pythagorean theorem applies to right triangle

ABH

formed by the altitude.

Account for the obtuse angle condition to find segment

CH

Because angle

ABC

is obtuse, the altitude foot

H

lies on the extension of segment

CB

beyond vertex

B

. Therefore,

CH = CB + BH = 25 + 9 = 34

For an obtuse triangle, the altitude to one of the adjacent sides falls outside the triangle.

Calculate side length

AC

using right triangle

AHC

AC = \sqrt{AH^2 + CH^2} = \sqrt{12^2 + 34^2} = \sqrt{144 + 1156} = \sqrt{1300} = 10\sqrt{13}

Applying the Pythagorean theorem to right triangle

AHC

yields the hypotenuse

AC

Key Concept

Triangle Area, Altitude Projection, and Extended Pythagorean Theorem

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