Question

In the following figure DE\(||\)BC, then: If DE = 4 cm, BC = 8 cm, A(\(\triangle\)ADE) = 25 cm\(^2\), find A(\(\triangle\)ABC).

Hint:

A common mistake is forgetting to square the ratio of the sides when dealing with the ratio of areas. Remember: side ratio is \(k\), area ratio is \(k^2\), and volume ratio (for 3D shapes) is \(k^3\).

Answer 1

Step 1: Understanding the Concept:
When a line parallel to one side of a triangle intersects the other two sides, it creates a smaller triangle that is similar to the original triangle. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Step 2: Key Formula or Approach:
1. First, prove \(\triangle\)ADE \(\sim\) \(\triangle\)ABC. 2. Then, use the theorem on areas of similar triangles: \[ \frac{A(\triangle ADE)}{A(\triangle ABC)} = \left(\frac{DE}{BC}\right)^2 = \left(\frac{AD}{AB}\right)^2 = \left(\frac{AE}{AC}\right)^2 \]

Step 3: Detailed Explanation:
Given: \[\begin{array}{rl} \bullet & \text{DE \(||\) BC} \\ \bullet & \text{DE = 4 cm, BC = 8 cm} \\ \bullet & \text{Area(\(\triangle\)ADE) = 25 cm\(^2\)} \\ \end{array}\] In \(\triangle\)ADE and \(\triangle\)ABC: \[\begin{array}{rl} \bullet & \text{\(\angle\)ADE = \(\angle\)ABC (Corresponding angles, since DE \(||\) BC)} \\ \bullet & \text{\(\angle\)AED = \(\angle\)ACB (Corresponding angles, since DE \(||\) BC)} \\ \bullet & \text{\(\angle\)DAE = \(\angle\)BAC (Common angle)} \\ \end{array}\] Therefore, by AA similarity criterion, \(\triangle\)ADE \(\sim\) \(\triangle\)ABC.
Now, using the theorem on the areas of similar triangles: \[ \frac{A(\triangle ADE)}{A(\triangle ABC)} = \left(\frac{DE}{BC}\right)^2 \] Substitute the given values: \[ \frac{25}{A(\triangle ABC)} = \left(\frac{4}{8}\right)^2 \] \[ \frac{25}{A(\triangle ABC)} = \left(\frac{1}{2}\right)^2 \] \[ \frac{25}{A(\triangle ABC)} = \frac{1}{4} \] Cross-multiply to solve for A(\(\triangle\)ABC): \[ A(\triangle ABC) = 25 \times 4 \] \[ A(\triangle ABC) = 100 \text{ cm}^2 \]

Step 4: Final Answer:
The area of \(\triangle\)ABC is 100 cm\(^2\).