Question

ABCD is a parallelogram. AP and CQ are perpendicular to diagonal DB. Prove that AP = CQ.

Answer 1

Given, ABCD is a parallelogram and AP and CQ are perpendiculars drawn from A and C to diagonal DB respectively.

To Prove: AP = CQ

Proof:

1. In parallelogram ABCD, diagonals bisect each other; let O be the midpoint of diagonal DB.
2. Since AP and CQ are perpendiculars on DB, AP ⊥ DB and CQ ⊥ DB.
3. Consider triangles △APD and △CQB.
4. AD = BC (Opposite sides of parallelogram are equal).
5. ∠APD = ∠CQB = 90° (Given, both perpendiculars).
6. PD = QB (Because O is midpoint of DB and P, Q lie perpendicular to DB; the segments PD and QB are segments on DB).
7. By RHS (Right angle-Hypotenuse-Side) congruence criterion, △APD ≅ △CQB.
8. Therefore, by CPCT (corresponding parts of congruent triangles), AP = CQ.

Hence, the perpendicular distances from A and C to diagonal DB are equal, so AP = CQ.