## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 11 Mid Point Theorem

**Question 1.**

**(a) In the figure (1) given below, D, E and F are mid-points of the sides BC, CA and AB respectively of ∆ ABC. If AB = 6 cm, BC = 4.8 cm and CA= 5.6 cm, find the perimeter of (i) the trapezium FBCE (ii) the triangle DEF.**

**(b) In the figure (2) given below, D and E are mid-points of the sides AB and AC respectively. If BC =**

**5.6 cm and∠B = 72°, compute (i) DE (ii)∠ADE.**

**(c) In the figure (3) given below, D and E are mid-points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram. Calculate AC if AF = 2.6 cm.**

**Solution:**

**(a) (i) Given :** AB = 6 cm, BC = 4.8 cm, and CA = 5.6 cm

**Required :** The perimeter of trapezium FBCA.

**Question 2.**

**Prove that the four triangles formed by joining in pairs the mid-points of the sides c of a triangle are congruent to each other.**

**Solution:**

**Given:** In ∆ ABC, D, E and r,

F are mid-points of AB, BC and CA respectively. Join DE, EF and FD.

**Question 3.**

**If D, E and F are mid-points of sides AB, BC and CA respectively of an isosceles triangle ABC, prove that ∆DEF is also F, isosceles.**

**Solution:**

**Given :** ABC is an isosceles triangle in which AB = AC

**Question 4.**

**The diagonals AC and BD of a parallelogram ABCD intersect at O. If P is the mid-point of AD, prove that**

**(i) PQ || AB**

**(ii) PO=\(\frac { 1 }{ 2 }\)CD.**

**Solution:**

**(i) Given :** ABCD is a parallelogram in which diagonals AC and BD intersect each other. At point O, P is the mid-point of AD. Join OP.

**Question 5.**

**In the adjoining figure, ABCD is a quadrilateral in which P, Q, R and S are mid-points of AB, BC, CD and DA respectively. AC is its diagonal. Show that**

**(i) SR || AC and SR =\(\frac { 1 }{ 2 }\)AC**

**(ii) PQ = SR**

**(iii) PQRS is a parallelogram.**

**Solution:**

**Question 6.**

**Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square, is also a square,**

**Solution:**

**Question 7.**

**In the adjoining figure, AD and BE are medians of ∆ABC. If DF U BE, prove that CF = \(\frac { 1 }{ 4 }\) AC.**

**Solution:**

**Question 8.**

**(a) In the figure (1) given below, ABCD is a parallelogram. E and F are mid-points of the sides AB and CO respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that**

**(i) ∆HEB = ∆HCF**

**(ii) GEHF is a parallelogram.**

**(b) In the diagram (2) given below, ABCD is a parallelogram. E is mid-point of CD and P is a point on AC such that PC = \(\frac { 1 }{ 4 }\) AC. EP produced meets BC at F. Prove that**

**(i) F is mid-point of BC (ii) 2EF = BD**

**Solution:**

**Question 9.**

**ABC is an isosceles triangle with AB = AC. D, E and F are mid-points of the sides BC, AB and AC respectively. Prove that the line segment AD is perpendicular to EF and is bisected by it.**

**Solution:**

**Question 10.**

**(a) In the quadrilateral (1) given below, AB || DC, E and F are mid-points of AD and BD respectively. Prove that:**

**Solution:**

**Question 11.**

**(a) In the quadrilateral (1) given below, AD = BC, P, Q, R and S are mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.**

**(b) In the figure (2) given below, ABCD is a kite in which BC = CD, AB = AD, E, F, G are mid-points of CD, BC and AB respectively. Prove that:**

**(i) ∠EFG = 90**

**(ii) The line drawn through G and parallel to FE bisects DA.**

**Solution:**

**Question 12.**

**In the adjoining figure, the lines l, m and n are parallel to each other, and G is mid-point of CD. Calculate:**

**(i) BG if AD = 6 cm**

**(ii) CF if GE = 2.3 cm**

**(iii) AB if BC = 2.4 cm**

**(iv) ED if FD = 4.4 cm.**

**Solution:**

**Multiple Choice Questions**

**Choose the correct answer from the given four options (1 to 6):**

**Question 1.**

**In a ∆ABC, AB = 3 cm, BC = 4 cm and CA = 5 cm. IfD and E are mid-points of AB and BC respectively, then the length of DE is**

**(a) 1.5 cm**

**(b) 2 cm**

**(c) 2.5 cm**

**(d) 3.5 cm**

**Solution:**

In ∆ABC, D and E are the mid-points of sides AB and BC

**Question 2.**

**In the given figure, ABCD is a rectangle in which AB = 6 cm and AD = 8 cm. If P and Q are mid-points of the sides BC and CD respectively, then the length of PQ is**

**(a) 7 cm**

**(b) 5 cm**

**(c) 4 cm**

**(d) 3 cm**

**Solution:**

**Question 3.**

**D and E are mid-points of the sides AB and AC of ∆ABC and O is any point on the side BC. O is joined to A. If P and Q are mid-points of OB and OC respectively, then DEQP is**

**(a) a square**

**(b) a rectangle**

**(c) a rhombus**

**(d) a parallelogram**

**Solution:**

D and E are mid-points of sides AB and AC respectively of AABC O is any point on BC and AO is joined P and Q are mid-points of OB and OC, EQ and DP are joined

**Question 4.**

**The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectanlge if**

**(a) PQRS is a parallelogram**

**(b) PQRS is a rectangle**

**(c) the diagonals of PQRS are perpendicular to each other**

**(d) the diagonals of PQRS are equal.**

**Solution:**

A, B, C and D are the mid-points of the sides PQ, QR, RS and SP respectively

**Question 5.**

**The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a rhombus if**

**(a) ABCD is a parallelogram**

**(b) ABCD is a rhombus**

**(c) the diagonals of ABCD are equal**

**(d) the diagonals of ABCD are perpendicular to each other.**

**Solution:**

P, Q, R and S are the mid-points of the quadrilateral ABCD and a quadrilateral is formed by joining the mid-points in order

**Question 6.**

**The figure formed by joining the mid points of the sides of a quadrilateral**

**ABCD, taken in order, is a square only if**

**(a) ABCD is a rhombus r**

**(b) diagonals of ABCD are equal**

**(c) diagonals of ABCD are perpendicular to each other**

**(d) diagonals of ABCD are equal and perpendicular to each other.**

**Solution:**

**Chapter Test**

**Question 1.**

**ABCD is a rhombus with P, Q and R as midpoints of AB, BC and CD respectively. Prove that PQ ⊥ QR.**

**Solution:**

**Question 2.**

**The diagonals of a quadrilateral ABCD are perpendicular. Show that the quadrilateral formed by joining the mid-points of its adjacent sides is a rectangle.**

**Solution:**

**Question 3.**

**If D, E, F are mid-points of the sides BC, CA and AB respectively of a ∆ ABC, Prove that AD and FE bisect each other.**

**Solution:**

**Question 4.**

**In ∆ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.**

**Solution:**

In ∆ABC, D and E are the mid-points of

**Question 5.**

**In the given figure, ABCD is a parallelogram and E is mid-point of AD. DL EB meets AB produced at F. Prove that B is mid-point of AF and EB = LF.**

**Solution:**

Given In the figure

**Question 6.**

**In the given figure, ABCD is a parallelogram. If P and Q are mid-points of sides CD and BC respectively. Show that CR = \(\frac { 1 }{ 2 }\) AC.**

**Solution:**