ML Aggarwal Solution Class 9 Chapter 11 Mid Point Theorem MCQS

  MCQS

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

Sol :

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

∴DE$=\frac{1}{2}$AC

But AC=5 cm

∴DE$=\frac{1}{2}\times 5=\frac{5}{2}$ cm =2.5cm

Ans (c)

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

Sol :

In the given figure

ABCD is a rectangle AB=6 cm , AD=8 cm

P and Q are mid points of BC and CD

∴PQ$=\frac{1}{2}$BD

But BD$=\sqrt{BC^2+CD^2}$ (pythagoras theorem)

$=\sqrt{8^2+6^2}$

$=\sqrt{64+36}=\sqrt{100}$

=10 cm

∴PQ$=\frac{1}{2}$BD

$=\frac{1}{2}\times 10=5$ cm

Ans (b)

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

Sol :

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

∵DE$=\frac{1}{2}BC $ and DE||BC..(i)

∴P and Q are mid points of BO and OC

∴PQ=PO+OQ

$=\frac{1}{2}$BO+$\frac{1}{2}$OC

$=\frac{1}{2}$(BO+OC)

$=\frac{1}{2}$(BC)..(ii)

=DE

From EQPD is a parallelogram 

Question 4

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle 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.

Sol :

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

The quadrilateral so formed by joining the mid points A, B , C , D is ABCD

ABCD will be rectangle , if the diagonals of PQRS bisects each other

i.e. PR and QS bisects each other 

Ans (c)

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.

Sol :

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

PQRS will be rhombus if the diagonals of ABCD are equal

i.e. AC=BD 

Ans (c)

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.

Sol :

P,Q,R and S are the mid points of the quadrilateral ABCD and a quadrilateral is formed by joining them in order. The quadrilateral so formed will be a square if the diagonals of ABCD are equal and perpendicular to each other.

i.e. AC and BD are equal and bisect it perpendicular 

Ans (d)