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In the provided image, we see a geometric problem that seems to involve triangles and trapezoids. The problem includes labeled dimensions for various segments of the figure.

Let’s break down the problem based on what we can observe:

- There is a right triangle \(\triangle ABC\) and a trapezoid.
- \(AB = 18 \, \text{cm}\), and the height from \(A\) to the base \(BC\) is marked as 20 cm.
- The base of the trapezoid, \(CD\), is given as 12 cm.
- There are two tasks:
  1. Calculate the area of triangle \(\triangle ABC\).
  2. Solve for the shaded area in the figure.

### (a) **Calculating the area of \(\triangle ABC\)**:
To calculate the area of a triangle, use the formula:

\[
\text{Area of } \triangle ABC = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, the base is \(AB = 18 \, \text{cm}\), and the height is 20 cm. Substituting these values:

\[
\text{Area of } \triangle ABC = \frac{1}{2} \times 18 \, \text{cm} \times 20 \, \text{cm} = 180 \, \text{cm}^2
\]

### (b) **Finding the shaded area**:
The shaded area seems to be the area of the trapezoid minus the triangle's area. We need to calculate the area of the trapezoid and subtract the area of the small triangle.

#### Trapezoid area:
For a trapezoid, the area is given by:

\[
\text{Area of trapezoid} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}
\]

- The parallel sides are \(AB = 18 \, \text{cm}\) and \(CD = 12 \, \text{cm}\).
- The height of the trapezoid is 20 cm (as seen from the figure).

So, the area of the trapezoid is:

\[
\text{Area of trapezoid} = \frac{1}{2} \times (18 \, \text{cm} + 12 \, \text{cm}) \times 20 \, \text{cm} = \frac{1}{2} \times 30 \, \text{cm} \times 20 \, \text{cm} = 300 \, \text{cm}^2
\]

#### Shaded area:
The shaded area is the difference between the trapezoid area and the triangle's area:

\[
\text{Shaded area} = 300 \, \text{cm}^2 - 180 \, \text{cm}^2 = 120 \, \text{cm}^2
\]

### Conclusion:
- The area of triangle \(\triangle ABC\) is \(180 \, \text{cm}^2\).
- The shaded area is \(120 \, \text{cm}^2\).

Would you like further details on the calculation steps?

Here are some related questions:
1. What formula is used to find the area of a triangle in general?
2. How do you calculate the area of a trapezoid?
3. Can you explain why we subtract areas when dealing with complex shapes?
4. What is the importance of height in calculating the areas of polygons?
5. How can you extend this method to solve for areas of other composite shapes?

**Tip:** Always ensure to verify the units when performing area calculations to avoid confusion between length and area units!

The problem provided in the image involves calculating the area of a triangle and determining the shaded area in a trapezoid.

This problem explores the calculation of the area of a right triangle and a trapezoid. It also covers finding the shaded area by subtracting the triangle's area from the trapezoid's total area. Ideal for students in Grades 7-9.

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