# Mid Point Theorem ML Aggarwal ICSE Class-9 Chapter-11

Mid Point Theorem ML Aggarwal ICSE Class-9 Chapter-11. Step by Step Answer of Exercise-11.1, Exercise-11.2, Exercise-11.3, MCQ and Chapter-Test Questions of  Mid Point Theorem  for ICSE Class-9 Mathematics. Visit official website CISCE for detail information about ICSE Board Class-9.

## Mid Point Theorem ML Aggarwal ICSE Class-9 Chapter-11

–: Select Topics :–

Exercise-11

MCQ

Chapter-Test ,

Note:- Before viewing Solution of Chapter -11 Mid Point Theorem Class-9 of ML Aggarwal Solutions. Read the Chapter Carefully then solve all example given in Exercise-11.

### Important point of  Mid Point Theorem

Mid Point Theorem is another properties of triangle but there is much difference in properties. Converse of Mid Point Theorem is also important for this chapter.Hence student should know about the basic of Mid Point Theorem. Theorem on equal intercept is an another properties on Mid Point Theorem .Chapter test is also given so that student can achieve their goal in exam of council. Practice of other book ( ICSE Publication )is also useful for clear the topic on Mid Point Theorem.

## Exercise-11  Mid Point Theorem ML Aggarwal

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.

(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.

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.

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.

(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.

#### Question 6.

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

#### Question 7.

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

#### 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.

#### 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.

#### Question 10.

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

#### 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.

#### 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.

### MCQ Mid Point Theorem ML Aggarwal Class-9

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

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

#### 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

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.

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.

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.

### Chapter Test Solutions of Mid Point Theorem for ML Aggarwal Class-9

#### Question 1.

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

#### 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.

#### 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.

#### 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.

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.

#### 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.

–: End of Mid Point Theorem :–