9) The measures of the diagonals of a rhombus are one $\frac{3}{4}$ of the other and their sum is $56 \mathrm{~m}$. Calculate the perimeter of the rhombus.
(A) $60 \mathrm{~m}$,
(B) $80 \mathrm{~m}$,
(C) $96 \mathrm{~m}$,
(D) $100 \mathrm{~m}$
(E) $108 \mathrm{~m}$.
9. The answer is $\mathbf{( B )}$.

$\overline{D B}+\overline{A C}=\frac{3}{4} \overline{A C}+\overline{A C}=\frac{7}{4} \overline{A C}=56 \mathrm{~m}$, that is $\overline{A C}=32 \mathrm{~m} \mathrm{and} \overline{B D}=24 \mathrm{~m}$.

The side of the rhombus is $\overline{A B}=\frac{1}{2} \cdot \sqrt{\overline{A C}^{2}+\overline{B D}^{2}}=\sqrt{16^{2}+12^{2}} \mathrm{~m}=20 \mathrm{~m}$. Consequently, the perimeter of the rhombus is $80 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_4701f996ee50400fb9c5g-2.jpg?height=340&width=280&top_left_y=2371&top_left_x=1613)