ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Solution:
We will use the mid-point theorem here. It states that the line segment joining the mid-points of any two sides of the triangle is parallel to the third side and is half of it.
Let us join AC and BD. In △ABC,
P and Q are the mid-points of AB and BC respectively.
∴ PQ || AC and PQ = 1/2 AC (Mid-point theorem) ... (1)
Similarly, in △ADC,
SR || AC and SR = 1/2 AC (Mid-point theorem) ... (2)
Clearly, PQ || SR and PQ = SR [From equation (1) and (2)]
Since in quadrilateral PQRS, one pair of opposite sides are equal and parallel to each other, it is a parallelogram.
∴ PS || QR and PS = QR (Opposite sides of the parallelogram) ... (3)
In △BCD, Q and R are the mid-points of side BC and CD respectively.
∴ QR || BD and QR = 1/2 BD (Mid-point theorem) ... (4)
However, the diagonals of a rectangle are equal.
∴ AC = BD ... (5)
Thus, QR = 1/2 AC
Also, in △BAD
PS || BD and PS = 1/2 BD
Thus, QR = PS .... (6)
By using Equations (1), (2), (3), (4), and (5), we obtain
PQ = QR = SR = PS
Therefore, PQRS is a rhombus.
☛ Check: NCERT Solutions Class 9 Maths Chapter 8
Video Solution:
ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus
NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.2 Question 3
Summary:
If ABCD is a rectangle and P, Q, R, S are mid-points of the sides AB, BC, CD, and DA respectively, then the quadrilateral PQRS is a rhombus.
☛ Related Questions:
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