Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.
Solution:
Given, two lines l and m intersect at the point O
P is a point on a line n passing through the point O such that P is equidistant from l and m.
P meets the line l at Q and the line m at R
Given, PQ = PR
The angle formed by l and m is ∠QOR.
We have to prove that n is the bisector of ∠QOR.
Considering triangles OQP and ORP,
Given, P is equidistant from the lines l and m.
So, PQ and PR should be perpendicular to the lines l and m.
∠PQO = ∠PRO = 90°
Common side = OP
Given, PQ = PR
RHS Congruence Rule states that two right triangles are congruent if the hypotenuse and one side of one triangle are equal to the corresponding hypotenuse and one side of the other triangle.
By RHS criterion, the triangles OQP and ORP are congruent
By CPCTC,
∠POQ = ∠POR
Therefore, n is the bisector of ∠QOR.
✦ Try This: In right angled triangle ABC, right angled at C,M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM=CM. Point D is joined to point B. Show that: △DBC≅△ACB
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7
NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 15
Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m
Summary:
Two lines l and m intersect at the point O and P is a point on a line n passing through the point O such that P is equidistant from l and m. It is proven that n is the bisector of the angle formed by l and m by CPCT
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