What is length of diagonal of the central 'square?

In square SQUA, the length of diagonal QA is 18 cm. In rhombus RHOM, the length of diagonal MH is 18 cm, and t

• In square SQUA, the length of diagonal QA is 18 cm. In rhombus RHOM, the length of diagonal MH is 18 cm, and the measure of angle RHO = 60 degrees. Which figure has the larger perimeter? What is the difference between the perimeters?

• Answer:

  Hi, in th square SQUA, let x be the length of each side then (SQ)^2 + (SA)^2 = (QA)^2, where SQ and SA are sides and QA is the diagonal SQ = SA = x cm (sides of square are equal) 2x^2 = 18^2 x = 12.73cm perimeter of square SQUA = 4x = 50.9 cm In the rhombus RHOM, <RHO = 60 degrees and diaginal MH = 18cm The diagonals MH and RO bisect each other perpendicularly at a point N. So MN = NH = 9cm Further the diagonal MH bisects the angle RHO into two angles of 30 degrees each. so <RHM or <RHN = 30 degrees In triangle RHN, <RNH = 90 degrees, < RHN = 30degrees and NH = 9 cm cos(RHN) = Adj/Hyp = NH/RH cos(30) = 9/RH RH = 9/cos(30) = 10.39 cm Perimeter of rhombus = 4*RH = 4 *10.39 = 41.6 cm Square SQUA has bigger perimeter than Rhombus RHOM and the diff. in perimeters = 50.1 - 41.6 = 8.5 cm Good luck