A rectangle PQRS has length PQ = 16 cm and width PS = 10 cm. Points X and Y are on sides PS and QR respectively such that PX = 4 cm and RY = 4 cm. Calculate the area of the shaded region PQRYX.

The length of the rectangle PQ is $16 \text{ cm}$.
The width of the rectangle PS is $10 \text{ cm}$.

$$\text{Area of rectangle PQRS} = \text{length} \times \text{width}$$ $$\text{Area of rectangle PQRS} = 16 \text{ cm} \times 10 \text{ cm} = 160 \text{ cm}^2$$

The unshaded triangles are $\triangle PQX$, $\triangle QRY$, and $\triangle SXY$.

• Area of $\triangle PQX$: This is a right-angled triangle at P. Base PQ = $16 \text{ cm}$. Height PX = $4 \text{ cm}$.

  $$\text{Area}(\triangle PQX) = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \text{ cm} \times 4 \text{ cm} = 32 \text{ cm}^2$$

• Area of $\triangle QRY$: This is a right-angled triangle at R. Base YR = $4 \text{ cm}$. Height QR = $10 \text{ cm}$.

  $$\text{Area}(\triangle QRY) = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \text{ cm} \times 10 \text{ cm} = 20 \text{ cm}^2$$

• Area of $\triangle SXY$: This is a right-angled triangle at S. First, find the lengths of SX and SY. SX = PS - PX = $10 \text{ cm} - 4 \text{ cm} = 6 \text{ cm}$. SY = SR - YR = $16 \text{ cm} - 4 \text{ cm} = 12 \text{ cm}$.

  $$\text{Area}(\triangle SXY) = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \text{ cm} \times 12 \text{ cm} = 36 \text{ cm}^2$$

$$\text{Total unshaded area} = \text{Area}(\triangle PQX) + \text{Area}(\triangle QRY) + \text{Area}(\triangle SXY)$$ $$\text{Total unshaded area} = 32 \text{ cm}^2 + 20 \text{ cm}^2 + 36 \text{ cm}^2 = 88 \text{ cm}^2$$

$$\text{Area of shaded part} = \text{Area of rectangle PQRS} - \text{Total unshaded area}$$ $$\text{Area of shaded part} = 160 \text{ cm}^2 - 88 \text{ cm}^2 = 72 \text{ cm}^2$$

The area of the shaded part is $\boxed{\text{72 cm}^2}$.

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