If the perimeter of a rhombus is 164 and the length of the longer diagonal is 80, find the length of the shorter diagonal.

A rhombus has four equal sides. If the perimeter of this rhombus is 164, then the length of one side is 164/4, or 41.

Draw this rhombus. Label all four sides with "41." Label the longer diagonal 80 and the half length of that diagonal 40. You will see inside the rhombus four congruent triangles with hypotenuse 41, leg 10 and unknown height. Thus, this unknown height is found by solving x^2 + 40^2 = 41^2, and x^2=9, so that the length of the shorter diagonal is 2 (2) = 18 (answer).