Discussion
In a rhombus ABCD, the length of the diagonals AC and BD are 16 cm and 12 cm, respectively. What is the perimeter (in cm)?
Answer: C
The perimeter is 40 cm. The diagonals of a rhombus bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 8 cm and 6 cm, half of each diagonal.
\[ \text{side} = \sqrt{8^{2} + 6^{2}} = \sqrt{100} = 10 \]\[ \text{perimeter} = 4 \times 10 = 40 \text{ cm} \]The other options are not four times 10 cm.
Check by the area: the area is half the product of the diagonals, 16 × 12 / 2 = 96 square cm, and the side is 10 cm, so the height on a side is 96 / 10 = 9.6 cm. This is consistent with a rhombus of side 10 cm, so the perimeter is 4 × 10 = 40 cm.
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