In the diagram shown, point D lies on side AB of triangle ABC, and point E lies on side AC… — SAT

In the diagram shown, point $D$ lies on side $AB$ of triangle $ABC$ , and point $E$ lies on side $AC$ . The lengths of the segments are $AB = 20$ , $AC = 16$ , $AD = 8$ , and $AE = 10$ . If the area of quadrilateral $BCED$ is $54$ , what is the area of triangle $ADE$ ?

(Note: Figure not drawn to scale.)

Answer: 18

Answer

Triangles ADE and ACB share angle A. The ratios of the adjacent sides of angle A are AD/AC = 8/16 = 1/2 and AE/AB = 10/20 = 1/2. By the Side-Angle-Side (SAS) similarity theorem, triangle ADE is similar to triangle ACB. The ratio of their areas is equal to the square of their similarity ratio: (1/2)^2 = 1/4. Therefore, the area of triangle ACB is 4 times the area of triangle ADE. The area of quadrilateral BCED is the difference between the area of triangle ACB and the area of triangle ADE, which is 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE). Since the area of quadrilateral BCED is 54, we have 3 * Area(ADE) = 54, which simplifies to Area(ADE) = 18.

Step-by-Step Solution

Calculate side ratios to establish similarity.

The ratio of AD to AC is 8/16 = 1/2, and the ratio of AE to AB is 10/20 = 1/2.

Checking if the corresponding sides surrounding the shared angle are in the same proportion.

Apply the Side-Angle-Side (SAS) similarity theorem.

Triangle ADE is similar to triangle ACB (triangle ADE ~ triangle ACB), where vertex A corresponds to A, D corresponds to C, and E corresponds to B.

Since the ratio of two pairs of corresponding sides is equal and their included angle is congruent, the triangles are similar.

Determine the area ratio based on the similarity scale factor.

The ratio of the area of triangle ADE to the area of triangle ACB is (1/2)^2 = 1/4.

The ratio of the areas of two similar figures is equal to the square of their similarity ratio.

Relate the area of the quadrilateral to the area of the smaller triangle.

Area(BCED) = Area(ACB) - Area(ADE) = 4 * Area(ADE) - Area(ADE) = 3 * Area(ADE).

The area of the quadrilateral is the difference between the areas of the larger and smaller triangles.

Solve for the area of triangle ADE.

Area(ADE) = 54 / 3 = 18.

Dividing the given area of the quadrilateral by 3 yields the area of the smaller triangle.

Key Concept

SAS Triangle Similarity and the Area Ratios of Similar Triangles

Estimated Time:2m 30s

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