Question

Points J, K, L, and M are midpoints of the sides of the rectangle EFGH. Prove that quadrilateral JKLM is a rhombus by finding the lengths of the sides. The diagram is not drawn to scale.

Answer 1

Distance from J to F = b
D from F to K = a
a^2+b^2=JK^2

D from K to G = a
D from G to L = b
a^2+b^2=KL^2

D from L to H = b
D from H to M = a
a^2+b^2=LM^2

D from M to E = a
D from E to J = b
a^2+b^2=MJ^2

For each side, I used the Pythagorean theorem (a^2+b^2=c^2) to find the length. Since every side of the quadrilateral squared (aka to the power of two) equals a^2+b^2, every side squared equals each other. So JK^2=KL^2=LM^2=MJ^2. If you take the square root of each side of the equal signs, you’re left with JK=KL=LM=MJ. In order for a quadrilateral to be a rhombus, each side must be equivalent. Each side in this quadrilateral is equivalent, therefore it is a rhombus.