Question

If DE : BC = 3 : 5, then find A(\(\triangle\)ADE) : A(\(\square\)DBCE).

Hint:

When asked for the ratio of the smaller triangle to the resulting trapezium, first find the ratio of the smaller triangle to the larger triangle (\(a^2/b^2\)). The ratio for the trapezium will then be (\(b^2-a^2\)). So the final ratio is \(a^2 : (b^2-a^2)\).

Answer 1

Step 1: Understanding the Concept:
As established in the previous question, since DE \(||\) BC, \(\triangle\)ADE \(\sim\) \(\triangle\)ABC. We can find the ratio of their areas using the given ratio of their sides. The area of the trapezium DBCE is the difference between the area of the larger triangle (\(\triangle\)ABC) and the smaller triangle (\(\triangle\)ADE).

Step 2: Key Formula or Approach:
1. \( \frac{A(\triangle ADE)}{A(\triangle ABC)} = \left(\frac{DE}{BC}\right)^2 \) 2. \( A(\square DBCE) = A(\triangle ABC) - A(\triangle ADE) \)

Step 3: Detailed Explanation:
Given: \[ \frac{DE}{BC} = \frac{3}{5} \] Since \(\triangle\)ADE \(\sim\) \(\triangle\)ABC, we have the ratio of their areas: \[ \frac{A(\triangle ADE)}{A(\triangle ABC)} = \left(\frac{DE}{BC}\right)^2 = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \] This means that if the area of \(\triangle\)ADE is 9k for some constant k, then the area of \(\triangle\)ABC is 25k.
Now, let's find the area of the trapezium DBCE. \[ A(\square DBCE) = A(\triangle ABC) - A(\triangle ADE) \] \[ A(\square DBCE) = 25k - 9k = 16k \] We need to find the ratio of A(\(\triangle\)ADE) to A(\(\square\)DBCE). \[ \frac{A(\triangle ADE)}{A(\square DBCE)} = \frac{9k}{16k} = \frac{9}{16} \]

Step 4: Final Answer:
The ratio A(\(\triangle\)ADE) : A(\(\square\)DBCE) is 9 : 16.