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