In acute triangle ABC, point D lies on side BC such that segment AD is perpendicular to BC… — GRE General Test

In acute triangle $ABC$ , point $D$ lies on side $BC$ such that segment $AD$ is perpendicular to $BC$ . The length of side $BC$ is $15$ , and the ratio of the area of triangle $ABD$ to the area of triangle $ADC$ is $2 : 3$ . If the lengths of sides $AB$ and $AC$ are both integers, what is the perimeter of triangle $ABC$ ?

A
$30$
B
$45$
C
$54$
$60$ Answer
E
$75$

Answer

The perimeter of triangle

ABC

60

The correct answer is derived by recognizing that the altitude splits the base into segments of lengths 6 and 9 based on the area ratio 2:3. Applying the Pythagorean theorem to both right triangles yields

AC^2 - AB^2 = 45

. Factoring 45 into positive integer pairs shows that the only valid side lengths yielding a real, positive height are

AB = 22

and

AC = 23

, giving a total perimeter of

15 + 22 + 23 = 60

Step-by-Step Solution

Determine the lengths of base segments

BD

and

DC

BD = 6

and

DC = 9

Triangles

ABD

and

ADC

share the common height

AD = h

. The ratio of their areas is equal to the ratio of their bases:

\frac{\text{Area}(ABD)}{\text{Area}(ADC)} = \frac{BD}{DC} = \frac{2}{3}

. Since

BD + DC = 15

, we have

BD = 6

and

DC = 9

Express the square of height

h^2

using the Pythagorean theorem in right triangles

ABD

and

ADC

h^2 = AB^2 - 36 = AC^2 - 81

In right triangle

ABD

AB^2 = BD^2 + h^2 = 36 + h^2

. In right triangle

ADC

AC^2 = DC^2 + h^2 = 81 + h^2

Set up a difference of squares equation for side lengths

AB

and

AC

(AC - AB)(AC + AB) = 45

Equating the two expressions for

h^2

gives

AC^2 - 81 = AB^2 - 36 \implies AC^2 - AB^2 = 45

Find positive integer solutions for

AB

and

AC

AB = 22

and

AC = 23

Since

AB

and

AC

are positive integers and

AC > AB

, we analyze factor pairs

(AC - AB, AC + AB)

45

with same parity (both odd):
- Pair

(1, 45)

AC - AB = 1

and

AC + AB = 45 \implies AC = 23, AB = 22

. Here

h^2 = 22^2 - 36 = 448 > 0

, giving a valid non-degenerate triangle.
- Pair

(3, 15)

AC - AB = 3

and

AC + AB = 15 \implies AC = 9, AB = 6

. Here

h^2 = 6^2 - 36 = 0

, which means

h = 0

(degenerate line segment, invalid).
- Pair

(5, 9)

AC - AB = 5

and

AC + AB = 9 \implies AC = 7, AB = 2

. Here

h^2 = 2^2 - 36 = -32 < 0

(impossible).

Calculate the total perimeter of triangle

ABC

Perimeter

= 15 + 22 + 23 = 60

Summing all three side lengths gives

BC + AB + AC = 15 + 22 + 23 = 60

Key Concept

Triangles: Properties, Perimeter, and Area

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