Question

In a parallelogram ABCD, the bisector of ∠A also bisects BC at X. Find AB : AD.

Answer 1

Parallelogram ABCD is given as follows:

We have AX bisects  ∠A bisecting BC at X.

That is,  BX = CX

We need to find AB :AD

Since, AX is the bisector ∠A

That is,

 `∠1 =1/2 ∠A ` …… (i)

Also, ABCD is a parallelogram

Therefore, AD || BC and AB intersects them

 ∠A +∠B = 180°

 ∠B = 180° - ∠A …… (ii)

In ΔABX,by angle sum property of a triangle:

∠1 +∠2 + ∠B = 180° 

From (i) and (ii), we get:

`1/2 ∠A + ∠2 + 180° - ∠A = 180°`

`∠2- 1/2 ∠A = 0` 

`∠2 = 1/2 ∠A` …… (iii)

From (i) and (iii),we get:

∠1 = ∠2 

Sides opposite to equal angles are equal. Therefore,

BX = AB

2BX = 2AB

As X is the mid point of BC. Therefore,

BC = 2AB

Also, ABCD is a parallelogram, then,  BC = AD

AD = 2AB

Thus,

AB :AD = AB :2AB

AB :AD = 1:2

Hence the ratio of AB : AD is 1:2.