Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, form a rectangle.
Solution:
Consider a rhombus ABCD
The points P, Q, R and S are the midpoints of the sides AB, BC, CD and AD.
We have to show that PQRS is a rectangle.
Join the diagonals AC and BD of the rhombus ABCD.
Considering the triangle ABD,
Since S and P are the midpoints of the sides AD and AB.
SP || BD ----------------- (1)
SP = 1/2 BD --------------- (2)
Similarly, RQ || BD
RQ = 1/2 BD -------------- (3)
From (2) and (3),
SP = RQ
Also, SP || RQ
Therefore, PQRS is a parallelogram
We know that the diagonals of a rhombus are perpendicular.
So, AC⊥ BD -------------- (4)
Considering triangle BAC,
PQ || AC --------------- (5)
From (1), (4) and (5),
SP ⊥ PQ
i.e.,∠SPQ = 90°
We know that a rectangle is a quadrilateral with four right angles. The opposite sides are parallel and equal to each other.
Therefore, PQRS is a rectangle.
✦ Try This: Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a rectangle is a rhombus.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.4 Sample Problem 3
Show that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, form a rectangle.
Summary:
It is shown that the quadrilateral formed by joining the mid-points the sides of a rhombus, taken in order, forms a rectangle as the angle is equal to 90 degrees
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