42 The diagonals of a rhombus are 16 cm and 30 cm long. What is the perimeter of the rhombus? \( \begin{array}{lll}\text { A } 68 \mathrm{~cm} & \text { B } 72 \mathrm{~cm} & \text { C } 80 \mathrm{~cm} \\ \text { D } 88 \mathrm{~cm} & \text { E } 92 \mathrm{~cm} & \end{array} \)

To find the perimeter of the rhombus, we first need to determine the length of one side. The diagonals of the rhombus bisect each other at right angles. Therefore, we can create right triangles using half of each diagonal. The lengths of the halves of the diagonals are: - Half of the first diagonal: \( 16 \, \text{cm} / 2 = 8 \, \text{cm} \) - Half of the second diagonal: \( 30 \, \text{cm} / 2 = 15 \, \text{cm} \) Using the Pythagorean theorem: \[ \text{side}^2 = (8 \, \text{cm})^2 + (15 \, \text{cm})^2 = 64 \, \text{cm}^2 + 225 \, \text{cm}^2 = 289 \, \text{cm}^2 \] Thus, the length of one side is: \[ \text{side} = \sqrt{289} = 17 \, \text{cm} \] The perimeter of the rhombus, which has four equal sides, is: \[ \text{Perimeter} = 4 \times 17 \, \text{cm} = 68 \, \text{cm} \] So the answer is A) 68 cm. Finding the perimeter of a rhombus can sometimes be underestimated, especially when you're calculating the side lengths. A common mistake is forgetting that the diagonals bisect each other and form right triangles. Stay sharp on these properties, and you'll ace those geometrical challenges! Additionally, the unique property of a rhombus can be linked to various real-world applications like design and architecture. Many structures utilize the rhombus shape for aesthetic appeal and stability. So next time you see a diamond-shaped window or ceiling design, remember that geometry is working behind the scenes to create beauty and strength!