If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.
Solution:
Given, ABC is a triangle
P, Q and R are the midpoints of the sides BC, CA and AB
AD is the perpendicular from A on BC
We have to prove that P, Q, R and D are concyclic.
Join DR, RQ and QP.
Consider right angled triangle ADP,
R is the midpoint of AB
So, RB = RD
We know that the angles opposite to equal sides are equal.
∠2 = ∠1 --------------------------------- (1)
Since R and Q are the midpoints of AB and AC,
The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.
By midpoint theorem,
RQ || BC
RQ || BP
We know that the opposite sides of a parallelogram are parallel and congruent.
Since QP || RB, BPQR is a parallelogram
We know that the opposite angles of a parallelogram are equal.
∠1 = ∠3 ------------------------------- (2)
From (1) and (2),
∠2 = ∠3 ------------------------------ (3)
We know that the linear pair of angles is equal to 180 degrees.
∠2 + ∠4 = 180
From (3),
∠3 + ∠4 = 180
We know that the pair of opposite angles of a cyclic quadrilateral is equal to 180 degrees.
Therefore, the points P, Q, R and D are concyclic.
✦ Try This: In a circle of radius 5cm, AB and CD are two parallel chords of lengths 8cm and 6cm respectively. Calculate the distance between the chords if they are on the opposite side of the centre.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 3
If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, prove that P, Q, R and D are concyclic.
Summary:
If P, Q and R are the mid-points of the sides BC, CA and AB of a triangle and AD is the perpendicular from A on BC, it is proven that P, Q, R and D are concyclic
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