Question

ABCD is a quadrilateral; P and Q are the midpoints of AB and DC, respectively. Prove that AD + BC = 2PQ.

Answer 1

In quadrilateral ABCD, if P and Q are the midpoints of sides AB and DC respectively, then the sum of AD and BC is equal to twice the length of PQ.

Step-by-step explanation:

In quadrilateral ABCD, let P and Q be the midpoints of sides AB and DC, respectively. We want to prove that AD + BC = 2PQ.

Since P is the midpoint of AB, we know that AP = PB. Similarly, since Q is the midpoint of DC, we know that CQ = QD.

Now, if we add AP + CQ, we get AD because AD = AP + PD. Similarly, if we add PB + QD, we get BC because BC = DC + CQ.

Therefore, AD + BC = (AP + PD) + (DC + CQ) = (AP + PB) + (CQ + QD) = 2PQ.