Area Composite Shape 59E122

1. **State the problem:** We need to find the area of a composite shape made of rectangle ABDE and right-angled triangle BCD. 2. **Identify given dimensions:** - Rectangle ABDE has width $4$ cm and height $9$ cm. - Triangle BCD is right-angled at D, with side BC = $15$ cm. 3. **Calculate the area of rectangle ABDE:** $$\text{Area}_{\text{rectangle}} = \text{width} \times \text{height} = 4 \times 9 = 36 \text{ cm}^2$$ 4. **Find the dimensions of triangle BCD:** - Since the triangle is right-angled at D, sides BD and CD are perpendicular. - The base of the triangle is BD, which equals the height of the rectangle, $9$ cm (since B and D are vertically aligned). - The hypotenuse BC is $15$ cm. 5. **Calculate the length of side CD using the Pythagorean theorem:** $$BC^2 = BD^2 + CD^2$$ $$15^2 = 9^2 + CD^2$$ $$225 = 81 + CD^2$$ $$CD^2 = 225 - 81 = 144$$ $$CD = \sqrt{144} = 12 \text{ cm}$$ 6. **Calculate the area of triangle BCD:** $$\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 9 \times 12 = 54 \text{ cm}^2$$ 7. **Calculate the total area of the shape:** $$\text{Area}_{\text{total}} = \text{Area}_{\text{rectangle}} + \text{Area}_{\text{triangle}} = 36 + 54 = 90 \text{ cm}^2$$ **Final answer:** The area of the shape is $90$ cm$^2$.