Quadiletrals and the properties of their diagonals
Q-1 ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig). AC is a diagonal. Show that:
and
PQRS is a parallelogram.
ABCD Is a Quadrilateral
ABCD is a quadrilateral such that P, Q, R, S are mid-point of the sides AB, BC, CD and DA
Solution:
Give, ABCD is a quadrilateral
P, Q, R and S are mid-points of the sides AB, BC, CD and DA
In ,
R is the midpoint of DC and S is the midpoint of DA.
Thus by mid-point theorem, and
In,
P is the midpoint of AB and Q is the midpoint of BC.
Thus by mid-point theorem, and
Also,
Thus,
- From (i)
And, - from (ii)
- From (i) and (ii)
Also,
Thus, PQRS is a parallelogram.
Q-2 ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
Solution:
ABCD Is Rhombus and PQRS Is Rectangle
ABCD is rhombus and PQRS is rectangle , P, Q, R, S are the mid-points of the side AB,BC,CD and DA respectively.
Given,
ABCD is a rhombus
P, Q, R and S are the mid-points of the sides AB, BC, CD and DA respectively.
Construction, AC and BD are joined.
Proof,
In and ,
(Halves of the opposite sides of the rhombus)
(Opposite angles of the rhombus)
(Halves of the opposite sides of the rhombus)
Thus, by Side-Angle-Side congruence condition.
By Corresponding Parts of Congruent Triangles ……equation (1)
In ΔQCR and ΔSAP,
(Halves of the opposite sides of the rhombus)
(Opposite angles of the rhombus)
(Halves of the opposite sides of the rhombus)
Thus, by Side-Angle-Side congruence condition.
By Corresponding Parts of Congruent Triangles ……equation (2)
Now, on
R and Q are the mid points of CD and BC respectively
Also,
P and S are the mid points of AD and AB respectively.
