P is a point on the bisector of ∠ABC. If the line through P, parallel to BA meet BC at Q, prove that BPQ is an isosceles triangle.

Solution:

Given, P is a point on the bisector ∠ABC.

The line through P, parallel to BA meets BC at Q.

We have to prove that BPQ is an isosceles triangle.

From the figure,

We observe that BP is the bisector of angle B.

∠ABP = ∠PBC

So, ∠1 = ∠2 ----------------- (1)

From the figure, PQ || AB

We know that the alternate interior angles are equal, if the two lines are parallel.

∠ABP = ∠BPQ

So, ∠1 = ∠3 ----------------- (2)

From (1) and (2),

∠2 = ∠3

We know that the sides opposite to equal angles are equal.

PQ = BQ

Therefore, BPQ is an isosceles triangle.

✦ Try This: In the given figure, BC = c;AC = a, AB = b and Altitude = p. Then:
(a) a² + b² = p²q²
(b) ab = p²
(c) 1/a² + 1/b² = 1/p²
(d) None of these

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7

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NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 4

P is a point on the bisector of ∠ABC. If the line through P, parallel to BA meet BC at Q, prove that BPQ is an isosceles triangle

Summary:

P is a point on the bisector of ∠ABC. If the line through P, parallel to BA meets BC at Q, it is proven that BPQ is an isosceles triangle. An Isosceles triangle is a triangle that has two equal sides

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