ABCD is a quadrilateral such that AB=AD and CB= CD. Prove that AC is the perpendicular bisector of BD

Question

Answer

As given , let ABCD be a quadrilateral wherein AB=AD and CB=CD.To prove :  AC is the perpendicular bisector of BD Consider triangle ADB , as AB=AD, it is an isosceles triangle.=> by property of isosceles triangle , angle ADB = angle ABD  therefore, triangle ADB is similar to triangle ABD. Now of similar triangle => side OD = side OB.  => AB/OB = AD/OD=> AO is a bisector of BD.  similarly, in triangle BCD, BC= CD=> it is also isosceles triangle , therefore angle CDB=angle CBD.and hence , triangle CDB is similar to triangle CBD.=> side OB= side OD.=> CD/OD  = CB/OB=> CO is a bisector of BD. As OA and OC is the bisector of triangle ABD and triangle BCD respectively. Therfore AC is a bisector of BD and perpendicular to BD.Hence proved.SORRY THE ANSWER IS VERY BIG.....BUT ITS JUST TO EXPLAIN IN BETTER WAY 🙌CHEARS!!!

Answer

sry i can't prove it here by writing it and answer by sending photo is not available

Answer

Statement: In quadrilateral ABCD, where AB = AD and CB = CD, AC is the perpendicular bisector of BD.

Proof:

1. Draw the diagram:

Let's start by drawing quadrilateral ABCD.

B
*
/ \
/ \
/ \
/ \
/ \
/ \
A-------------C
\ /
\ /
\ /
\ /
\ /
\ /
D

2. Given information:

Given that AB = AD and CB = CD.

3. Define perpendicular bisector:

A perpendicular bisector is a line that divides a line segment into two equal parts and intersects it at a right angle.

4. Assume AC is the perpendicular bisector of BD:

Let's assume that AC is the perpendicular bisector of BD.

5. Prove AC bisects BD:

To prove that AC bisects BD, we need to show that AC divides BD into two equal parts and intersects it at a right angle.

6. Prove AC divides BD into two equal parts:

Since AB = AD, triangles ABC and ADC are congruent by the Side-Side-Side (SSS) congruence criterion. Therefore, angle ABC = angle ADC.

Similarly, since CB = CD, triangles CBA and CDA are congruent by the Side-Side-Side (SSS) congruence criterion. Therefore, angle CBA = angle CDA.

Since angle ABC = angle ADC and angle CBA = angle CDA, we can conclude that angle ABD = angle ADB.

Therefore, AC divides BD into two equal parts.

7. Prove AC intersects BD at a right angle:

To prove that AC intersects BD at a right angle, we need to show that angle BAC = angle DAC.

Since AB = AD, triangles ABC and ADC are congruent by the Side-Side-Side (SSS) congruence criterion. Therefore, angle BAC = angle DAC.

Therefore, AC intersects BD at a right angle.

8. Conclusion:

Since AC divides BD into two equal parts and intersects it at a right angle, we can conclude that AC is the perpendicular bisector of BD.

Therefore, it is proven that in quadrilateral ABCD, where AB = AD and CB = CD, AC is the perpendicular bisector of BD.

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