Question

Question 9 In the given figure, D is a point on side BC of ΔABC such that. Prove that AD is the bisector of ∠BAC.

Class 10 - Math - Triangles Page 153

Answer 1

Basic proportionality theorem

If a line is drawn parallel to one side of a triangle to intersect the other two side interesting points than the other two sides are divided in the same ratio and this theorem is also known as Thales theorem.

Converse of basic proportionality theorem

If a line divides any two sides of a triangle in the same ratio then the line must parallel to the third side.

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Figure is in the attachment.

Solution:

Given:
D is a point on side BC of ∆ABC, such that

BD/CD= AB/AC

To Prove: AD is the bisector of ∠BAC.

Construction: BA is produced to E, such that AE= AC.
Join CE.

Proof:
BD/CD= AB/AC. (Given)

BD/CD= AB/AE
[AC= AE(by construction)]

AD || CE

[ by Converse of BPT]

In ∆BCE,
∠BAD= ∠AEC....................(i)
[Corresponding angles]

∠CAD = ∠ACE..................(ii)
[Alternate interior angles]

AC= AE. ( by construction)

∠AEC= ∠ACE..................(iii)
[Since the angles opposite to equal sides of a triangle are equal]

From eq i, ii, iii

∠BAD = ∠CAD

Hence, AD is the bisector of∠BAC.
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