## RS Aggarwal Class 9 Solutions Chapter 9 Quadrilaterals and Parallelograms Ex 9C

These Solutions are part of RS Aggarwal Solutions Class 9. Here we have given RS Aggarwal Class 9 Solutions Chapter 9 Quadrilaterals and Parallelograms Ex 9C.

**Other Exercises**

- RS Aggarwal Solutions Class 9 Chapter 9 Quadrilaterals and Parallelograms Ex 9A
- RS Aggarwal Solutions Class 9 Chapter 9 Quadrilaterals and Parallelograms Ex 9B
- RS Aggarwal Solutions Class 9 Chapter 9 Quadrilaterals and Parallelograms Ex 9C

**Question 1.**

**In the adjoining figure, ABCD is a trapezium in which AB || DC and E is the midpoint of AD. A line segment EF || AB meets BC at F. Show that F is the midpoint of BC.**

**Solution:**

Given : In trapezium ABCD,

AB || DC and E is the midpoint of AD.

A line EF ||AB is drawn meeting BC at F.

To prove : F is midpoint of BC

**Question 2.**

**In the adjoining figure, ABCD is a ||gm in which E and F are the midpoints of AB and CD respectively. If GH is a line segment that cuts AD, EF and BC at G, P and H respectively, prove that GP = PH.**

**Solution:**

Given : In ||gm ABCD, E and F are the mid points of AB and CD respectively. A line segment GH is drawn which intersects AD, EF and BC at G, P and H respectively.

To prove : GP = PH

**Question 3.**

**In the adjoining figure, ABCD is a trapezium in which AB || DC and P, Q are the midpoints of AD and BC respectively. DQ and AB when produced meet at E. Also, AC and PQ intersect at R.**

**prove that (i) DQ = QE, (ii) PR || AB, (iii) AR = RC. .**

**Solution:**

Given : In trapezium ABCD, AB || DC

P, Q are the midpoints of sides AD and BC respectively

DQ is joined and produced to meet AB produced at E

Join AC which intersects PQ at R.

**Question 4.**

**In the adjoining figure, AD is a median of ∆ ABC and DE || BA. Show that BE is also a median of ∆ ABC.**

**Solution:**

Given : In ∆ ABC,

AD is the mid point of BC

DE || AB is drawn. BE is joined.

To prove : BE is the median of ∆ ABC.

Proof : In ∆ ABC

**Question 5.**

**In the adjoining figure, AD and BE are the medians of ∆ ABC and DF || BE. Show that CF = BC.**

**Solution:**

Given : In ∆ ABC, AD and BE are the medians. DF || BE is drawn meeting AC at F.

To prove : CF = BC.

**Question 6.**

**In the adjoining figure, ABCD is a parallelogram. E is the mid point of DC and through D, a line segment is drawn parallel to EB to meet CB produced at G and it cuts AB at F.**

**Prove that (i) AD = GC,**

**(ii) DG = 2EB.**

**Solution:**

Given : In ||gm ABCD, E is mid point of DC.

EB is joined and through D, DEG || EB is drawn which meets CB produced at G and cuts AB at F.

To prove : (i)AD = GC

**Question 7.**

**Prove that the line segments joining the middle points of the sides of a triangle divide it into four congruent triangles.**

**Solution:**

Given : In ∆ ABC,

D, E and F are the mid points of sides BC, CA and AB respectively

DE, EF and FD are joined.

**Question 8.**

**In the adjoining figure, D, E, F are the midpoints of the sides BC, CA and AB respectively, of A ABC. Show that ∠EDF = ∠A, ∠DEF = ∠B and ∠DEF = ∠C**

**Solution:**

Given : In ∆ ABC, D, E and F are the mid points of sides BC, CA and AB respectively

**Question 9.**

**Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rectangle is a rhombus.**

**Solution:**

Given : In rectangle ABCD, P, Q, R and S are the midpoints of its sides AB, BC, CD and DA respectively PQ, QR, RS and SP are joined.

To prove : PQRS is a rhombus.

**Question 10.**

**Show that die quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a rhombus is a rectangle.**

**Solution:**

Given : In rhombus ABCD, P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively PQ, QR, RS and SP are joined.

**Question 11.**

**Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides of a square is a square.**

**Solution:**

Given : In square ABCD, P,Q,R and S are the mid points of sides AB, BC, CD and DA respectively. PQ, QR, RS and SP are joined.

**Question 12.**

**Prove that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.**

**Solution:**

Given : In quadrilateral ABCD, P, Q, R and S are the midpoints of PQ, QR, RS and SP respectively PR and QS are joined.

To prove : PR and QS bisect each other

Const. Join PQ, QR, RS and SP and AC

Proof : In ∆ ABC,

But diagonals of a ||gm bisect each other PR and QS bisect each other.

**Question 13.**

**In the given figure, ABCD is a quadrilateral whose diagonals intersect at right angles. Show that the quadrilateral formed by joining the midpoints of the pairs of adjacent sides is a rectangle.**

**Solution:**

Given : ABCD is a quadrilateral. Whose diagonals AC and BD intersect each other at O at right angles.

P, Q, R and S are the mid points of sides AB, BC, CD and DA respectively. PQ, QR, QS and SP are joined.

Hope given RS Aggarwal Class 9 Solutions Chapter 9 Quadrilaterals and Parallelograms Ex 9C are helpful to complete your math homework.

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