Question

In a triangle ABC, a line \(\overline{DE}\) is drawn parallel to \(\overline{BC}\) such that D lies on \(\overline{AB}\) and E lies on \(\overline{AC}\). If \(\overline{AD}:\overline{DB} = 2:3\), find the ratio of the areas of triangle ADE and ABC.

• A. \(2:5\)
• B. \(4:25\)
• C. \(2:3\)
• D. \(4:9\)

Hint:

Key theorem: For similar triangles, \(\text{Area ratio} = (\text{side ratio})^2\). The most common mistake here is confusing \(AD:DB\) with \(AD:AB\). Always find the full side \(AB = AD + DB\) first, then compute the ratio of the partial side to the full side.

Answer 1

The Correct Option is B

Concept:
If two triangles are similar, the ratio of their areas equals the

square of the ratio of their corresponding sides:
By the Basic Proportionality Theorem (Thales' theorem), if \(DE \parallel BC\) and \(D\) is on \(AB\), \(E\) is on \(AC\), then \(\triangle ADE \sim \triangle ABC\) (AA similarity: angle A is common, and \(\angle ADE = \angle ABC\) since \(DE \parallel BC\) means corresponding angles are equal).

Step 1: Find \(AD:AB\) from the given ratio \(AD:DB = 2:3\).
Given \(AD:DB = 2:3\), this means:



Step 2: Confirm similarity of triangles.
Since \(DE \parallel BC\):

• \(\angle A\) is common to both \(\triangle ADE\) and \(\triangle ABC\).

• \(\angle ADE = \angle ABC\) (corresponding angles, since \(DE \parallel BC\)).

• \(\angle AED = \angle ACB\) (corresponding angles, since \(DE \parallel BC\)).
By AA similarity: \(\triangle ADE \sim \triangle ABC\).

Step 3: Apply the areas ratio theorem for similar triangles.


Step 4: Understand why other options are wrong.

• \(2:5\): This is the ratio of sides \(AD:AB\), not of areas. Areas use the square of the side ratio.

• \(2:3\): This is the given ratio \(AD:DB\), not the area ratio.

• \(4:9\): This would be the ratio if \(AD:DB = 2:3\) were used directly as side ratio (i.e.
if it were \(AD:AB = 2:3\) rather than \(AD:DB = 2:3\)).