A5. The lengths of the diagonals of a rhombus have a ratio of $3: 4$. (A rhombus is a quadrilateral with four sides of equal length.) The sum of the lengths of the diagonals is 56. What is the perimeter of the rhombus?
(A) 80
(B) 96
(C) 100
(D) 108
(E) 160
A5. The lengths of the diagonals of a rhombus have a ratio of $3: 4$. (A rhombus is a quadrilateral with four sides of equal length.) The sum of the lengths of the diagonals is 56. What is the perimeter of the rhombus? (A) 80

Solution The diagonals are $\frac{3}{7} \cdot 56=24$ and $\frac{4}{7} \cdot 56=32$. The half-diagonals are therefore $12(=4 \cdot 3)$ and 16 $(=4 \cdot 4)$. The rhombus is thus 4 times as large as the rhombus in the figure, which is composed of four 3-4-5 triangles. The side is therefore $4 \cdot 5=20$ and the perimeter is thus $4 \cdot 20=80$.

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