A rhombus has sides of length 51 units each and a shorter diagonal of length 48 units. What is the length, in units, of the longer diagonal?
I know that a rhombus is a quadrilateral with four congruent sides, and that its diagonals are perpendicular bisectors of each other. I can draw a rhombus and label its sides and diagonals. I can also divide the rhombus into four right triangles by drawing the diagonals. In each right triangle, the hypotenuse is 51 units, one leg is 24 units (half of the shorter diagonal), and the other leg is half of the longer diagonal. I can use the Pythagorean theorem to find the length of the longer diagonal. Let x be the length of the longer diagonal. Then x/2 is the leg of the right triangle, and I have (x/2)^2 + 24^2 = 51^2. Expanding and simplifying, I get x^2/4 + 576 = 2601. Multiplying by 4, I get x^2 + 2304 = 10404. Subtracting 2304, I get x^2 = 8100. Taking the square root, I get x = 90. The length of the longer diagonal is 90 units. 