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# 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 = 1/2AC

(ii) PQ = SR

(iii) PQRS is a parallelogram.

**Solution:**

We will use the mid-point theorem here. It that 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.

(i) In ΔADC, S and R are the mid-points of sides AD and CD respectively. Thus, by using the mid-point theorem

∴ SR || AC and SR = 1/2AC ... (1)

(ii) In ΔABC, P and Q are mid-points of sides AB and BC. Therefore, by using the mid-point theorem,

PQ || AC and PQ = 1/2 AC ... (2)

Using Equations (1) and (2), we obtain PQ || SR and PQ = SR ... (3)

∴ PQ = SR

(iii) From Equation (3), we obtained PQ || SR and PQ = SR

Clearly, one pair of opposite sides of quadrilateral PQRS is parallel and equal. Hence, PQRS is a parallelogram.

**☛ Check: **NCERT Solutions Class 9 Maths Chapter 8

**Video Solution:**

## 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 = 1/2AC (ii) PQ = SR (iii) PQRS is a parallelogram.

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.2 Question 1

**Summary:**

If ABCD is a quadrilateral in which P, Q, R, S are mid-points of the sides AB, BC, CD, and DA, AC is a diagonal, then SR || AC and SR = 1/2 AC, PQ = SR, and PQRS is a parallelogram.

**☛ Related Questions:**

- 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.
- 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.
- ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig. 8.30). Show that F is the mid-point of BC
- In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

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