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

**Solution:** --- ### **a) Finding the Length of the Second Rectangle** **Given:** - **First Rectangle Dimensions:** 21 cm (length) × 7 cm (width) - **Second Rectangle Width:** 3 cm - **Both Rectangles Have the Same Area** **Step 1: Calculate the Area of the First Rectangle** \[ \text{Area}_1 = \text{Length}_1 \times \text{Width}_1 = 21\ \text{cm} \times 7\ \text{cm} = 147\ \text{cm}^2 \] **Step 2: Set Up the Equation for the Second Rectangle** Let \( L \) be the length of the second rectangle. \[ \text{Area}_2 = \text{Width}_2 \times L = 3\ \text{cm} \times L \] Since the areas are equal: \[ 3\ \text{cm} \times L = 147\ \text{cm}^2 \] **Step 3: Solve for \( L \)** \[ L = \frac{147\ \text{cm}^2}{3\ \text{cm}} = 49\ \text{cm} \] **Answer:** The length of the second rectangle is **49 cm**. --- ### **b) Finding the Area of the Square** **Given:** - **Triangle Base:** 5 m - **Triangle Height:** 8 m - **Perimeter of the Square = 3 × Area of the Triangle** **Step 1: Calculate the Area of the Triangle** \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 5\ \text{m} \times 8\ \text{m} = 20\ \text{m}^2 \] **Step 2: Calculate the Perimeter of the Square** \[ \text{Perimeter}_{\text{square}} = 3 \times \text{Area}_{\text{triangle}} = 3 \times 20\ \text{m}^2 = 60\ \text{m} \] **Step 3: Find the Side Length of the Square** Let \( s \) be the side length of the square. \[ \text{Perimeter}_{\text{square}} = 4 \times s \\ 60\ \text{m} = 4s \\ s = \frac{60\ \text{m}}{4} = 15\ \text{m} \] **Step 4: Calculate the Area of the Square** \[ \text{Area}_{\text{square}} = s^2 = (15\ \text{m})^2 = 225\ \text{m}^2 \] **Answer:** The area of the square is **225 m²**. ---