In `DeltaABC, DE||BC` where D and E are respectively lie on AB and AC and `DE:BC=3:5`. What is the ratio of area of triangle ABC to area of triangle DAE ?

Options

A. `3:1`
B. `5:3`
C. `9:2`
D. `25:9`

Answer

To solve the problem, we need to find the ratio of the area of triangle ABC to the area of triangle DAE given that DE is parallel to BC and the ratio of DE to BC is 3:5.

### Step-by-step Solution:

1. **Identify the triangles and the given information**:
- We have triangle ABC with DE parallel to BC.
- Points D and E lie on sides AB and AC respectively.
- The ratio of DE to BC is given as \( \frac{DE}{BC} = \frac{3}{5} \).

2. **Establish the similarity of triangles**:
- Since DE is parallel to BC, by the Basic Proportionality Theorem (or Thales' theorem), triangle ADE is similar to triangle ABC.
- This gives us the relationship of their corresponding sides and angles:
- \( \angle A \) is common in both triangles.
- \( \angle ABC \) is equal to \( \angle ADE \) (corresponding angles).
- \( \angle ACB \) is equal to \( \angle AED \) (corresponding angles).
- Thus, by the Angle-Angle (AA) similarity criterion, we can conclude that \( \triangle ABC \sim \triangle ADE \).

3. **Use the ratio of corresponding sides**:
- From the similarity, we know that the ratio of the corresponding sides of the triangles is the same as the ratio of DE to BC.
- Therefore, \( \frac{AB}{AD} = \frac{AC}{AE} = \frac{BC}{DE} = \frac{5}{3} \).

4. **Calculate the ratio of areas**:
- The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
- Thus, the ratio of the areas of triangle ABC to triangle DAE is:
\[ \frac{Area \, of \, \triangle ABC}{Area \, of \, \triangle DAE} = \left(\frac{BC}{DE}\right)^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9}. \]

5. **Final answer**:
- Therefore, the ratio of the area of triangle ABC to the area of triangle DAE is \( \frac{25}{9} \).