Lestion 3 a) Two rectangles have the same area. If the first rectangle has dimensions 21 cm by 7 cm , and the second rectangle has a width of 3 cm , then find the length of the second rectangle. b) The perimeter of a square is thrice the area of a triangle. Given that the base of the triangle measures 5 m and the vertical height is 8 m , find the area of the square. [ 6 marks] Total: 25 marks

To find the length of the second rectangle, we can start by calculating the area of the first rectangle. The area is given by multiplying length by width: \(21 \text{ cm} \times 7 \text{ cm} = 147 \text{ cm}^2\). Since both rectangles share the same area, we set the equation for the second rectangle as \(3 \text{ cm} \times \text{length} = 147 \text{ cm}^2\). Solving for length gives us \( \text{length} = \frac{147 \text{ cm}^2}{3 \text{ cm}} = 49 \text{ cm}\). Now, for the area of the square, we first need to determine the area of the triangle using the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). Substituting the values we have: \( \text{Area} = \frac{1}{2} \times 5 \text{ m} \times 8 \text{ m} = 20 \text{ m}^2\). The perimeter of the square is three times the area of the triangle, so \( P = 3 \times 20 \text{ m}^2 = 60 \text{ m}\). Since the perimeter \( P = 4 \times \text{side}\), we find that \( \text{side} = \frac{60 \text{ m}}{4} = 15 \text{ m}\) and the area of the square is \( \text{Area} = \text{side}^2 = 15^2 = 225 \text{ m}^2\).