ABC is an isoceles triangle with base `BC=12 cm` . There is a rectangle GHED inside the triangle whose base is GH on side BC. HE=6cm, F is the midpoint of BC . If AF=24cm, then find area of rectangle.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step-by-Step Solution:

1. **Draw the Triangle and Rectangle**:
- Start by drawing triangle ABC, where BC is the base measuring 12 cm. Since ABC is an isosceles triangle, the lengths of sides AB and AC are equal.
- Mark point F as the midpoint of BC. Therefore, BF = FC = 6 cm.

2. **Identify Given Measurements**:
- The height from point A to line BC is given as AF = 24 cm.
- The height of the rectangle HE = 6 cm.

3. **Establish Similar Triangles**:
- Consider triangles CHE and CFA. Since AF is perpendicular to BC, both triangles will have a right angle at points H and F respectively.
- The angles at C in both triangles are equal (common angle), which makes triangles CHE and CFA similar.

4. **Set Up the Ratio**:
- Let FH = x. Then, CH = FC - FH = 6 - x.
- From the similarity of triangles, we can set up the ratio:
\[ \frac{CH}{FC} = \frac{HE}{AF} \]
- Substituting the known values gives:
\[ \frac{6 - x}{6} = \frac{6}{24} \]

5. **Cross-Multiply to Solve for x**:
- Cross-multiplying gives:
\[ 24(6 - x) = 6 \cdot 6 \]
- Simplifying this results in:
\[ 144 - 24x = 36 \]
- Rearranging gives:
\[ 24x = 144 - 36 \]
\[ 24x = 108 \]
\[ x = \frac{108}{24} = 4.5 \text{ cm} \]

6. **Calculate the Length of GH**:
- Since GH = FH + FH = 2x, we have:
\[ GH = 2 \times 4.5 = 9 \text{ cm} \]

7. **Calculate the Area of Rectangle GHED**:
- The area of the rectangle is given by:
\[ \text{Area} = GH \times HE = 9 \text{ cm} \times 6 \text{ cm} = 54 \text{ cm}^2 \]

### Final Answer:
The area of rectangle GHED is **54 cm²**.