Question

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 8.29). AC is a diagonal. Show that: (i) SR || AC and SR = 1/2 AC (ii) PQ = SR (iii) PQRS is a parallelogram.​

Answer 1

Step-by-step explanation:

The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

(i) In △DAC , S is the mid point of DA and R is the mid point of DC. Therefore, SR∥AC and SR=

2/

1

AC.By mid-point theorem.

(ii) In △BAC , P is the mid point of AB and Q is the mid point of BC. Therefore, PQ∥AC and PQ=

2/

1

AC.By mid-point theorem. But from (i) SR=

2/

1

AC therefore PQ=SR

(iii) PQ∥AC & SR∥AC therefore PQ∥SR and PQ=SR. Hence, a quadrilateral with opposite sides equal and paralle is a parallelogram. Therefore PQRS is a parallelogram.

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Answer 2

Step-by-step explanation:

Given

P, Q, R and S are the mid points of quadrilateral ABCD

Solution

Theorem :The line segment joining the mid-points of two sides of a triangle is parallel to the third side.

(i)  SR || AC and SR = 1/2 AC

Considering ∆ACD

we observe that S and R are the mid points of side AD and DC respectively. [Given]

Hence,SR || AC and SR = ½ AC (As per the above theorem)……………………….(1)

(ii) PQ = SR

Considering ∆ACB

We observe that P and Q are the mid points of side AB and BC respectively. [Given]

Hence, PQ || AC and PQ = ½ AC (As per above theorem)…………………………(2)

From (1) and (2) we can say,

PQ = SR

(iii) PQRS is a parallelogram

From (i) and (ii) we can say that

PQ || AC and SR || AC

so, PQ || SR and PQ = SR

If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Hence, PQRS is a parallelogram.

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