1. **State the problem:** We are given two rhombuses and need to find their perimeter ($P$) and area ($A$). 2. **Recall formulas for a rhombus:** - Perimeter: $P = 4 \times \text{side}$ - Area: $A = \text{base} \times \text{height}$ or $A = \frac{d_1 \times d_2}{2}$ where $d_1$ and $d_2$ are diagonals. --- ### Rhombus 3: - Side length = 10.7 ft - One diagonal = 8 ft - Diagonals intersect at right angles 3. **Find the other diagonal:** Since diagonals intersect at right angles and the rhombus has equal sides, use the Pythagorean theorem on half diagonals: Let the other diagonal be $d_2$. Half diagonals: $\frac{8}{2} = 4$ ft and $\frac{d_2}{2}$. Using Pythagoras: $$10.7^2 = 4^2 + \left(\frac{d_2}{2}\right)^2$$ $$114.49 = 16 + \frac{d_2^2}{4}$$ $$114.49 - 16 = \frac{d_2^2}{4}$$ $$98.49 = \frac{d_2^2}{4}$$ Multiply both sides by 4: $$4 \times 98.49 = d_2^2$$ $$393.96 = d_2^2$$ Take square root: $$d_2 = \sqrt{393.96} \approx 19.85 \text{ ft}$$ 4. **Calculate area:** $$A = \frac{d_1 \times d_2}{2} = \frac{8 \times 19.85}{2} = \frac{158.8}{2} = 79.4 \text{ ft}^2$$ 5. **Calculate perimeter:** $$P = 4 \times 10.7 = 42.8 \text{ ft}$$ --- ### Rhombus 5: - Side length = 19 m - Height = 9 m - Base = side = 19 m 6. **Calculate area:** $$A = \text{base} \times \text{height} = 19 \times 9 = 171 \text{ m}^2$$ 7. **Calculate perimeter:** $$P = 4 \times 19 = 76 \text{ m}$$ --- **Final answers:** - Rhombus 3: $P = 42.8$ ft, $A = 79.4$ ft$^2$ - Rhombus 5: $P = 76$ m, $A = 171$ m$^2$