Prove that a quadrilateral is a rectangle if and only if the

Prove that a quadrilateral is a rectangle if and only if the diagonals are congruent and bisect each other

Solution

If the diagonals of a quadrilateral bisect each other and have the same length, then the quadrilateral is a rectangle.

Proof: Since the diagonals bisect each other, we know that it is a parallelogram, so all we need to prove is that the angles at the vertices are right angles.

Again let the quadrilateral be ABCD with diagonals AC and BD intersecting at P. Since the diagonals bisect each other, P is the midpoint of both diagonals. That is, AP = PB and CP = PD. But the diagonals are also of the same length, so AP + PB = CP + PD, and by substitution this gives us AP + AP = CP + CP, or 2AP= 2CP. That is, AP = CP. Likewise, PB = PD. Consequently, all 4 triangles APD, APB, CPD, and BPC are isosceles. So angles PAD and PDAare congruent, angles PBC and PCB are congruent, angles PAB and PBA are congruent, and angles PDCand PCD are congruent. And the triangles APD and CPB are congruent, and the triangles APB and CPD are conguent. By angle addition, it follows that the 4 angles of the quadrilateral (angles ABC, BCD, CDA, and DAB) are all equal. But the angles of a quadrilateral add to 360o, and therefore each of these 4 angles must be 90°.

Hence the quadilateral is rectangle if and only if diagonals areconguent and bisect each other.