Question

Diagonal AC is the perpendicular bisector of diagonal BD in the quadrilateral ABCD.

Prove that

1. AB = AD 
2. BC = DC

Answer 1

Given: Diagonal AC is the perpendicular bisector of diagonal BD in the quadrilateral ABCD.

To Prove:

1. AB = AD 
2. BC = DC

Proof:

1. Since AC is the perpendicular bisector of BD, it means:

AC ⟂ BD   ...(AC is perpendicular to BD)

AC bisects BD, so O is the midpoint of BD such that BO = OD

2. Consider triangles △ABO and △ADO:

AO is common to both triangles.

BO = DO   ...(O is midpoint of BD)

∠AOB = ∠AOD = 90°   ...(Since AC is perpendicular to BD)

Therefore, by RHS (Right angle-Hypotenuse-Side) congruence, △ABO ≅ △ADO

3. From congruence, corresponding parts are equal:

AB = AD

4. Similarly, consider triangles △BCO and △DCO:

CO is common.

BO = DO   ...(Midpoint)

∠BCO = ∠DCO = 90°   ...(Perpendicularity)

Again by RHS congruence,

△BCO ≅ △DCO

5. From this congruence,

BC = DC

Hence proved:

1. AB = AD 
2. BC = DC