Question

ABCD is a rectangle and P,QR and S are mid point of the sides AB,BC,CD,DA are respectively. Show that the quadrilateral PQRS is a rhombus ????
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Answer 1

Answer:

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Step-by-step explanation:

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.  

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

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.