If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.
Solution:
Given, the bisector of an angle of a triangle also bisects the opposite side.
We have to prove that the triangle is isosceles.
Consider a triangle ABC with a point D on BC such that BD = DC
∠BAD = ∠CAD
Extend AD to meet E such that AD = DE and join CE.
Considering triangles ABD and ECD,
Given, BD = CD
Given, AD = DE
We know that the vertically opposite angles are equal.
∠ADB = ∠EDC
By SAS criterion, the triangles ABD and ECD are congruent.
In triangles ABD and ECD,
AB = EC ------------- (1)
∠BAD = ∠CED
Given, ∠BAD = ∠CAD
So, ∠CED = ∠CAD
We know that the sides opposite to the equal angles are equal.
So, AC = EC ---------- (2)
From (1) and (2),
AB = AC
Therefore, the triangle ABC is isosceles.
✦ Try This: In a ΔABC, AD is the bisector of ∠A, meeting side BC at D. If AC = 4.2 cm, DC = 6 cm and 10 cm, find AB
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Sample Problem 3
If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles
Summary:
If the bisector of an angle of a triangle also bisects the opposite side, it is proven that the triangle is isosceles. An isosceles triangle is a type of triangle that has any two sides equal in length
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