Question

O is a point inside the rectangle ABCD. Prove that OB² + OD² = OA² + OC².

Answer 1

Given: O is a point inside the rectangle ABCD.

To prove: OB² + OD² = OA² + OC².

Construction: Join O, with the vertices A, B, C and D of rectangle ABCD. 

Now draw  a parallel line LM through O which meets AB and DC at M and L respectively.

Proof: In ∆OMB

OB² = OM² + MB² = OM² + CL²

(by Baudhayan theorem)

Since, ABCD is a rectangle and ML ⊥ AB

MB = CL and in ∆ODL

OD² = OL² + DL²

= OL² + AM² (∵ DL = AM)

∴ OB² + OD² = OM² + CL² + OL² + AM²

= (OM² + AM²) + (CL² + OL²)

= OA² + OC²

Hence, OB² + OD² = OA² + OC²

Hence proved.