In the figure below not drawn to scale, AB = 28 cm. DE = 4 cm. CE = 16 cm. Find the area of the shaded part.

1. Break Down the Problem

To find the area of the shaded part, we need to determine the area of the larger triangle (formed by points A, B, and C) and the area of the smaller triangle (formed by points D, E, and C). The shaded area will be the area of the larger triangle minus the area of the smaller triangle.

2. Relevant Concepts

We can use the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

3. Analysis and Detail

• Step 1: Find the area of triangle ABC.
  • Given $AB = 28 \, \text{cm}$ (this will be the base).
  • The height can be determined as follows: We need to first find the length $AC$.

Assuming $AE = EC$ for simplicity, then:

$AE + EC = AC$

• Step 2: Find the area of triangle DEC.
  • Given $DE = 4 \, \text{cm}$ and $CE = 16 \, \text{cm}$.

Assuming the height from point E to base DC is the same as for both triangles, we can use the base lengths directly.

4. Verify and Summarize

First, we calculate the area of triangle ABC and then triangle DEC.

• Calculating Area of Triangle ABC: $\text{Area}_{ABC} = \frac{1}{2} \times AB \times h = \frac{1}{2} \times 28 \times h$

• Calculating Area of Triangle DEC: $\text{Area}_{DEC} = \frac{1}{2} \times DE \times h = \frac{1}{2} \times 4 \times h$

Now, subtract the area of triangle DEC from triangle ABC to get the shaded area:

$\text{Shaded Area} = \text{Area}_{ABC} - \text{Area}_{DEC}$

Final Answer

Assuming the height $h$ is the same for both triangles, the final calculation of the shaded area is:

$\text{Shaded Area} = \left(\frac{1}{2} \times 28 \times h\right) - \left(\frac{1}{2} \times 4 \times h\right)$

This simplifies to:

$\text{Shaded Area} = \frac{1}{2} \times (28 - 4) \times h = \frac{1}{2} \times 24 \times h = 12h$

Thus, the area of the shaded part is $12h$ square centimeters, dependent on the height $h$. If the height $h$ is known, substitute the value to get a numerical answer.