https://file.aiquickdraw.com/mathbot/file/1725099853485zj7oxpi0.jpg

To find the area of the quadrilateral \(PQRS\), we need to analyze the given image and use the appropriate geometric formulas.

### Analysis of the Shape

From the image:

- \(PQRS\) is a quadrilateral composed of two right triangles: \(\triangle PQS\) and \(\triangle QRS\).
- Given:
  - \(PQ = 20 \, \text{cm}\) (base of \(\triangle PQS\))
  - \(PS = 16 \, \text{cm}\) (hypotenuse of \(\triangle PQS\))
  - \(QR = 6 \, \text{cm}\) (base of \(\triangle QRS\))
  - Both triangles have right angles at \(Q\) and \(S\).

### Step-by-step Solution

1. **Finding the Height \(QS\):**
   - Triangle \(PQS\) is a right triangle with hypotenuse \(PS = 16 \, \text{cm}\) and base \(PQ = 20 \, \text{cm}\).
   - To find the height \(QS\), we use the Pythagorean theorem:
   
\[
PS^2 = PQ^2 + QS^2
\]

Substitute the values:

\[
16^2 = 20^2 + QS^2
\]

\[
256 = 400 + QS^2
\]

\[
QS^2 = 256 - 400
\]

\[
QS^2 = 144
\]

\[
QS = 12 \, \text{cm}
\]

2. **Finding the Area of Triangles \(PQS\) and \(QRS\):**

   - **Area of \(\triangle PQS\):**

\[
\text{Area}_{\triangle PQS} = \frac{1}{2} \times PQ \times QS
\]

Substitute the values:

\[
\text{Area}_{\triangle PQS} = \frac{1}{2} \times 20 \times 12 = 120 \, \text{cm}^2
\]

   - **Area of \(\triangle QRS\):**

To find the height of \(\triangle QRS\) from \(S\) to \(QR\), we notice that the triangle is also a right triangle with base \(QR = 6 \, \text{cm}\) and height \(QS = 12 \, \text{cm}\) (same as above).

\[
\text{Area}_{\triangle QRS} = \frac{1}{2} \times QR \times QS
\]

Substitute the values:

\[
\text{Area}_{\triangle QRS} = \frac{1}{2} \times 6 \times 12 = 36 \, \text{cm}^2
\]

3. **Calculating the Total Area of Quadrilateral \(PQRS\):**

\[
\text{Area}_{PQRS} = \text{Area}_{\triangle PQS} + \text{Area}_{\triangle QRS}
\]

\[
\text{Area}_{PQRS} = 120 + 36 = 156 \, \text{cm}^2
\]

### Conclusion

The area of quadrilateral \(PQRS\) is \(156 \, \text{cm}^2\).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the area of a quadrilateral with four given sides?
2. What other methods can you use to calculate the area of a non-standard quadrilateral?
3. How can the Pythagorean theorem help in finding areas of more complex shapes?
4. What is the significance of knowing the height in calculating the area of triangles?
5. How can coordinate geometry be used to solve for areas of irregular polygons?

**Tip:** When working with right triangles, always consider using the Pythagorean theorem or trigonometric ratios to find missing lengths, which can help solve for areas more accurately.

Learn how to calculate the area of quadrilateral PQRS using geometric formulas and the Pythagorean Theorem. This problem involves analyzing a quadrilateral composed of two right triangles, with given side lengths. Suitable for students in Grades 7-9.

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