Perimeter Area Similar Bb1A99

1. **State the problem:** We have two similar rectangles ABCD and WXYZ. - Rectangle ABCD has side BC = $11$ in and perimeter $66$ in. - Rectangle WXYZ has side XY = $66$ in and unknown perimeter. We need to find: (1) The perimeter of WXYZ. (2) Explanation of how the answer was found. (3) The area of WXYZ. 2. **Recall the properties of similar figures:** - Corresponding sides of similar figures are proportional. - The ratio of perimeters is equal to the ratio of corresponding sides. - The ratio of areas is the square of the ratio of corresponding sides. 3. **Find the scale factor:** The side BC corresponds to side XY. Scale factor $k = \frac{\text{side XY}}{\text{side BC}} = \frac{66}{11} = 6$ 4. **Find the perimeter of WXYZ:** Since perimeters scale by the same factor as sides, $$\text{Perimeter of WXYZ} = k \times \text{Perimeter of ABCD} = 6 \times 66 = 396$$ 5. **Explain how the answer was found:** We used the fact that similar figures have proportional sides and perimeters. The ratio of the sides was $6$, so the perimeter of the larger rectangle is $6$ times the smaller rectangle's perimeter. 6. **Find the area of WXYZ:** Let $A_1$ be the area of ABCD and $A_2$ be the area of WXYZ. The ratio of areas is the square of the scale factor: $$\frac{A_2}{A_1} = k^2 = 6^2 = 36$$ We need to find $A_1$ first. 7. **Find the area of ABCD:** Perimeter of ABCD is $66$ in. Perimeter formula for rectangle: $$P = 2(l + w)$$ Given one side $w = 11$ in, $$66 = 2(l + 11) \Rightarrow 33 = l + 11 \Rightarrow l = 22$$ Area of ABCD: $$A_1 = l \times w = 22 \times 11 = 242$$ 8. **Calculate area of WXYZ:** $$A_2 = 36 \times 242 = 8712$$ 9. **Explain how area is affected:** When a shape is scaled by a factor $k$, its area is scaled by $k^2$. Here, since the side length increased by $6$, the area increased by $6^2 = 36$ times.