### Instruction: The diagonals of a rhombus measure 18 feet and 12 feet. What is the perimeter of the rhombus? Express your answer in simplest radical form.### Response: I know that a rhombus is a parallelogram with four congruent sides, and that its diagonals bisect each other at right angles.
I can draw a picture of the rhombus and label the lengths of the diagonals.
I can see that the diagonals form four right triangles, each with legs of 9 feet and 6 feet, since they are half of the diagonals.
I can use the Pythagorean theorem to find the length of the hypotenuse of one of these triangles, which is also the length of one side of the rhombus.
I have a^2 + b^2 = c^2, where a = 9, b = 6, and c is the unknown side length.
Simplifying, I get 81 + 36 = c^2, or 117 = c^2.
Taking the square root of both sides, I get c = sqrt(117).
I can simplify the radical by factoring out the largest perfect square that divides 117, which is 9.
I get c = sqrt(9) * sqrt(13), or c = 3 * sqrt(13).
This is the length of one side of the rhombus, and since all four sides are congruent, the perimeter is four times this length.