In a rectangle the angle between diagonals is 60°. The sum of the lengths of both diagonals and both shorter sides of the rectangle is 36 in.

Question

What are the lengths of the diagonals?

Answer 1

As ALMOST ALWAYS IN GEOMETRY, you should make and label a diagram. In this case you can label the shorter sides as having length 2x EACH. Relying upon the 60^o angle at the intersection of diagonals, and drawing a line segment from that intersection to the midpoint of one of the shorter sides, you will have formed a 30-60-90 triangle whose SHORTER SIDE measures x. Its hypotenuse will measure 2x due to the properties of 30-60-90 triangles. This means that each diagonal measures 4x.

 

Two diagonals measure 8x

Two shorter sides measure 4x

Total of the above is 12x = 36 in so x = 3 in.  Therefore the diagonals measure 12 in.