If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic.

Solution:

Given, ABC is a triangle

BM and CN are the perpendiculars drawn on the sides AC and AB

We have to prove that B, C, M and N are concyclic.

Since BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC

∠BMC = ∠BNC = 90°

We know that if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, then the four points are concyclic.

Here, the line segment BC joining two points B and C subtends equal angles of 90 degrees at M and N.

Therefore, the four points B, C , M and N are concyclic.

✦ Try This: The radius of the circle passing through the center of the incircle of △ABC and through the end points of BC is given by

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10

NCERT Exemplar Class 9 Maths Exercise 10.3 Problem 10

If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, prove that the points B, C, M and N are concyclic.

Summary:

If BM and CN are the perpendiculars drawn on the sides AC and AB of the triangle ABC, it is proven that the points B, C, M and N are concyclic

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