In \triangle ABC, the lengths of sides AB, BC, and AC are 13, 14, and 15, respectively. A … — GRE General Test

In $\triangle ABC$ , the lengths of sides $AB$ , $BC$ , and $AC$ are $13$ , $14$ , and $15$ , respectively. A line segment $DE$ is drawn parallel to side $BC$ , with point $D$ lying on side $AB$ and point $E$ lying on side $AC$ . If the perimeter of $\triangle ADE$ is equal to the perimeter of quadrilateral $DBCE$ , what is the area of $\triangle ADE$ ?

Answer: 47.25

Answer

47.25

The area of the original triangle

\triangle ABC

is computed as

84

using Heron's formula. By defining the linear scale factor

k

between

\triangle ADE

and

\triangle ABC

, the perimeters of

\triangle ADE

and quadrilateral

DBCE

are expressed as

42k

and

42 - 14k

, respectively. Setting these equal yields

k = 0.75

. The area of

\triangle ADE

is then

k^2 \times 84 = 0.5625 \times 84 = 47.25

Step-by-Step Solution

Calculate the perimeter and area of the main triangle

\triangle ABC

The perimeter of

\triangle ABC

13 + 14 + 15 = 42

. Using Heron's formula with semi-perimeter

s = 21

\text{Area}(\triangle ABC) = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84

Finding the area and perimeter of the full triangle sets the required baseline for proportional scaling.

Set up expressions for the perimeters of

\triangle ADE

and quadrilateral

DBCE

using a scale factor

k

Because

DE \parallel BC

\triangle ADE \sim \triangle ABC

with scale factor

k = \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}

. Thus,

\text{Perimeter}(\triangle ADE) = 13k + 15k + 14k = 42k

. The segments

DB = 13(1-k)

and

EC = 15(1-k)

, so

\text{Perimeter}(DBCE) = 13(1-k) + 14 + 15(1-k) + 14k = 42 - 14k

Parallel lines create similar triangles, which allows all perimeter segment lengths to be represented in terms of one variable

k

Solve for the scale factor

k

by equating the two perimeters.

42k = 42 - 14k \implies 56k = 42 \implies k = \frac{42}{56} = \frac{3}{4} = 0.75

Equating the perimeters satisfies the condition specified in the question stem.

Calculate the area of

\triangle ADE

using the square of the linear scale factor.

\text{Area}(\triangle ADE) = k^2 \times \text{Area}(\triangle ABC) = \left(\frac{3}{4}\right)^2 \times 84 = \frac{9}{16} \times 84 = \frac{189}{4} = 47.25

The area ratio of similar geometric figures is proportional to the square of their linear scale factor.

Key Concept

Properties of Similar Triangles, Area via Heron's Formula, and Perimeter Scaling

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