P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.
Solution:
Given, ABCD is a quadrilateral
P, Q, R and S are the midpoints of the sides AB, BC, CD and AD
AC ⊥ BD
We have to prove that PQRS is a rectangle.
Since AC ⊥ BD
∠AOD = ∠AOB = ∠BOC = ∠COD = 90°
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.”
Considering triangle ADC,
S and R are the midpoints of AD and DC
By midpoint theorem,
SR || AC
SR = 1/2 AC --------------- (1)
Considering triangle ABC,
P and Q are the midpoints of AB and BC
By midpoint theorem,
PQ || AC
PQ = 1/2 AC ----------------- (2)
Comparing (1) and (2),
SR = PQ = 1/2 AC ------------ (3)
Considering triangle BAD,
SP || BD
By midpoint theorem,
SP = 1/2 BD ----------------- (5)
Comparing (4) and (5),
SP = RQ = 1/2 BD ----------- (6)
Considering quadrilateral EOFR,
OE || FR
OF || ER
∠EOF = ∠ERF = 90°
Therefore, PQRS is a rectangle.
✦ Try This: In △ABC, ∠A=50° and ∠B=60°. Determine the longest and the shortest sides of the triangle.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 4
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. Prove that PQRS is a rectangle.
Summary:
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC ⊥ BD. It is proven that PQRS is a rectangle
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