Question

ABCD is a quadrilateral in which AP and CQ are perpendicular to diagonal BD and AP = CQ. Prove that BD bisects AC.

Answer 1

Given:

ABCD is a quadrilateral in which AP and CQ are perpendicular to the diagonal BD.

AP = CQ

To Prove: BD bisects AC, meaning that M, the point of intersection of BD and AC, divides AC into two equal parts, i.e., AM = MC.

Proof:

1. Perpendiculars and Given Conditions:

From the given information, AP and CQ are perpendicular to BD, meaning:

∠APB = 90°

∠CQD = 90°

Also, AP = CQ  ...(Given)

2. Consider Triangles ΔABP and ΔCDQ:

We have the following information:

AP = CQ  ...(Given)

∠APB = ∠CQD = 90°  ...(Since both are perpendicular)

BP = DQ  ...(Since BD is a common side and the perpendiculars divide BD into equal segments)

Therefore, by the Hypotenuse-Leg (HL) Congruence Theorem (since both triangles are right-angled and share the leg BD and hypotenuses AP and CQ of equal length) ΔABP ≅ ΔCDQ.

3. Corresponding Parts of Congruent Triangles:

Since ΔABP ≅ ΔCDQ, their corresponding parts are equal:

AB = CD

BP = DQ

4. Point M is the Midpoint of AC:

Now, consider the intersection of diagonal BD with AC.

Since BD is a common side of congruent triangles ΔABP and ΔCDQ, the point of intersection M divides AC into two equal segments.

Specifically, since AB = CD, the line BD must bisect AC, meaning AM = MC.

Thus, BD bisects AC.