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In DeltaABC, D and E are points on the s...

In `DeltaABC`, D and E are points on the sides AB and AC respectively, such that DE || BC and `DE:BC=6:7`. (Area of `DeltaADE`) : (Area of trapezium BCED) = ?

A

`49:13`

B

`36:13`

C

`13:49`

D

`13:36`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the ratio of the area of triangle ADE to the area of trapezium BCED given that DE is parallel to BC and the ratio of their lengths is 6:7. ### Step-by-Step Solution: 1. **Understanding the Given Information:** - We have triangle ABC. - Points D and E are on sides AB and AC respectively. - DE is parallel to BC. - The ratio of the lengths DE to BC is given as 6:7. 2. **Using the Properties of Similar Triangles:** - Since DE is parallel to BC, triangles ADE and ABC are similar. - The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides. - Therefore, we can write: \[ \frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle ABC} = \left(\frac{DE}{BC}\right)^2 = \left(\frac{6}{7}\right)^2 = \frac{36}{49} \] 3. **Letting the Area of Triangle ABC be A:** - If we let the area of triangle ABC be A, then: \[ \text{Area of } \triangle ADE = \frac{36}{49} A \] 4. **Finding the Area of Trapezium BCED:** - The area of trapezium BCED can be found by subtracting the area of triangle ADE from the area of triangle ABC: \[ \text{Area of trapezium BCED} = \text{Area of } \triangle ABC - \text{Area of } \triangle ADE = A - \frac{36}{49} A = \frac{49}{49} A - \frac{36}{49} A = \frac{13}{49} A \] 5. **Finding the Ratio of Areas:** - Now we can find the ratio of the area of triangle ADE to the area of trapezium BCED: \[ \frac{\text{Area of } \triangle ADE}{\text{Area of trapezium BCED}} = \frac{\frac{36}{49} A}{\frac{13}{49} A} = \frac{36}{13} \] 6. **Final Answer:** - Thus, the ratio of the area of triangle ADE to the area of trapezium BCED is: \[ \text{Area of } \triangle ADE : \text{Area of trapezium BCED = } 36 : 13 \]
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