Page 10.34

### EXERCISE 10.2

**Type 1**

#### QUESTION 1

Fill in the blanks :

(i) In a ΔPQR , M and N are the mid-points of PQ and PR respectively , then MN=__

(ii) In a ΔABC , D and E are the mid-points of AB and AC respectively , if AB=7.4cm , BC=5.4cm and AC=4.4 cm , then length of DE will be __

(iii) In a ΔABC , D and E are the mid-points of the side AB and AC respectively. If DE=2.6 cm , then length of BC will be __

(iv) In a triangle , line joining the mid-points of two sides is parallel and __ of the third side

(v) If a line is divided by the three lines in the ratio 1:3 , then second line will be divided by these three lines in the ratio __

Sol :

#### QUESTION 2

**What is the relation between the line segment joining the mid-points of two sides and the third side of a triangle ?**

Sol :

#### QUESTION 3

**Three parallel lines AD , BE and CF are intersected by two transversals at points A, B , C and D , E , F . If AB=BC=4 cm , then find the value of DE:EF**

Sol :

#### QUESTION 4

**Which figure is formed when mid-points of consecutive sides of a quadrilateral are joined ?**

Sol :

#### QUESTION 5

**What type of triangle is formed when mid-points of consecutive sides of a equilateral triangle are joined ?**

Sol :

#### QUESTION 6

**In ΔDEF , A and B are the mid-points of the sides DE and DF , then what type of quadrilateral ABEF will be found ?**

Sol :

**TYPE 2**

#### QUESTION 7

**ABCD is a trapezium in which AB||CD and P and Q are the mid-points of AD and BC. If AB=4 cm , CD=7 cm , then find PQ**

[Hint : Line segment joining the mid-points of non-parallel sides of a trapezium is half of the sum of the parallel sides.]

Sol :

#### QUESTION 8

**In the adjoining figure, ΔABC is an equilateral triangle . On joining the mid-points of the sides of ΔABC , ΔDEF is formed. Again , on joining the mid-points of the sides of the ΔDEF , ΔPQR is formed. If AB=12 cm , then find the length of sides of the ΔPQR**

Sol :

#### QUESTION 9

**If ABCD is a trapezium in which AB||DC , E is the mid-point of AD and F is the mid-point of BC.**

**(i) If AB=4 cm and DC=6cm , then find EF**

**(ii) If AB=8cm and DC=6cm , then find EF**

Sol :

#### QUESTION 10

**In a trapezium ABCD , AB||DC and point E is the mid-point of the side AD. a line through the point E and parallel to the side AB meets the line BC in F . Prove that F is the mid-point of BC. [Hint : join AC]**

Sol :

**TYPE 3**

#### QUESTION 11

**In parallelogram ABCD , P and Q are the mid-points of DC and BC respectively. Prove that CR=1/2 AC**

[Hint : Join B and D and let BD and AC intersect at O . Now , OC=1/2AC. Now in ΔCBD , P and Q are the mid-points of DC and BC respectively

∴ PQ||BD or PR||DO and RQ||OB

Now in ΔOCD , PR||OD

∴ R is the mid-point of OC

Now CR=1/2 OC = 1/2 (1/2AC)=1/4 AC [∵ OC=1/2 AC]

]

Sol :

#### QUESTION 12

**In ΔABC, M and N are points on AB and AC , such that AM=1/4 AB and AN=1/4 AC . Prove that MN=1/4 BC**

[Hint: Let P and Q be the mid-points of AB and AC respectively

∴PQ=1/2 BC

Now , M and N are the mid-points of AP and AQ respectively\

∴MN=1/2 PQ = 1/4 BC]

Sol :

#### QUESTION 13

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

<fig to be added>

[Hint : AB=AC, BD=DC , AD=AD

⇒ ΔABD≅ΔACD [SSS congruence rule]

⇒∠ADB≅∠ADC=90°

⇒ F is the mid-point of AB and E is the mid-point of AC

⇒FE||BC

⇒ ∠AOF≅∠ADC=90° [corresponding angles]

⇒ F is the mid-point of AB and FO||BD

⇒ O is the mid-point of AD

Hence AD⊥EF and AO=OD]

Sol :

#### QUESTION 14

**In figure, three parallel lines l, m and n are intersected by the transversal p at points A , B and C respectively and by the transversal ‘q’ at D , E and F respectively.**

**If AB:BC=1:2 , then prove that DE:EF=1:2**

<fig to be added>

Sol :