In \triangle ABC, point D lies on side BC such that the ratio of the length of segment BD … — GRE General Test

In $\triangle ABC$ , point $D$ lies on side $BC$ such that the ratio of the length of segment $BD$ to the length of segment $DC$ is $3 : 2$ . Point $E$ is the midpoint of line segment $AD$ . If the area of $\triangle ABC$ is $60$ square units, which of the following statements must be true? Select all that apply.

The area of $\triangle ABD$ is $36$ square units.Answer
The area of $\triangle ABE$ is $18$ square units.Answer
The area of $\triangle BEC$ is $30$ square units.Answer
D
The area of $\triangle ADC$ is $30$ square units.
E
The area of $\triangle ECD$ is $15$ square units.

Answer

The statements asserting that the area of triangle ABD is 36 square units, the area of triangle ABE is 18 square units, and the area of triangle BEC is 30 square units are all true.

Triangles sharing a vertex and having bases along the same straight line share the same height. Thus, their areas are in the exact ratio of their bases. Since segment BD is 3/5 of BC, triangle ABD has an area of (3/5) * 60 = 36 square units. Median BE divides triangle ABD into two equal areas of 18 square units each. Median CE divides triangle ADC (area 24) into two equal areas of 12 square units each. Summing triangles EBD (18) and ECD (12) gives an area of 30 square units for triangle BEC.

Step-by-Step Solution

Determine the areas of triangles ABD and ADC using the base ratio.

Area of triangle ABD = 36 square units, and Area of triangle ADC = 24 square units.

Triangles ABD and ADC share the same altitude from vertex A to line BC. Therefore, their areas are directly proportional to their base lengths BD and DC. Since BD : DC = 3 : 2, BD is 3/5 of BC and DC is 2/5 of BC.

Calculate the area of triangle ABE.

Area of triangle ABE = 18 square units.

Point E is the midpoint of segment AD. In triangle ABD, line segment BE is a median from vertex B to side AD. A median bisects a triangle into two region of equal area, so Area(ABE) = 1/2 * Area(ABD) = 1/2 * 36 = 18 square units.

Calculate the area of triangle ECD.

Area of triangle ECD = 12 square units.

Similarly, segment CE is a median in triangle ADC from vertex C to side AD. Thus, Area(ECD) = 1/2 * Area(ADC) = 1/2 * 24 = 12 square units.

Calculate the area of triangle BEC.

Area of triangle BEC = 30 square units.

Triangle BEC is formed by combining triangles EBD and ECD. Since Area(EBD) = 18 and Area(ECD) = 12, Area(BEC) = 18 + 12 = 30 square units.

Key Concept

Triangles sharing a vertex and altitude have areas proportional to their bases; a median divides a triangle into two equal areas.

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