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**Exercise 6.2****Q.1. In figures (i) and (ii), DE y BC. Find EC in (i) and AD in (ii).****Sol.**

[By basic proportional theorem]

(ii) In ΔABC, DE || BC

[By basic proportional theorem]**Q.2. E and F are points on the sides PQ and PR respectively of a ΔPQR. For each of the following cases, state whether EF || QR:****(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm****(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm****(iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm****Sol.** (i)

⇒ EF is not parallel to QR

[By converse of B.P.T.]

(ii)

⇒ EF **||** QR [By converse of B.P.T.]

(iii)

⇒ EF **||** QR [By converse of B.P.T.]**Q.3. In the figure, if LM || CB and LN || CD, prove that **.

From equation (i) and (ii)**Q.4. In the figure, DE || AC and DF || AE. Prove that**

From equation (i) and (ii)

In ΔPOR, DF || OR

...(ii)

From equation (i) and (ii), we get

∴ EF || QR [By converse of B.P.T.]

AB || PQ

∴

In ΔPOR, AC || PR

∴ OP/PA = OR/RC .......(ii)

From equation (i) and (ii), we get

BC || QR [By converse of B.P.T.]**Q.7. Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. ****Sol.** Given: In Δ ABC, D is the mid-point of AB and DE || BC

To Prove: AE = EC

Proof: In ΔABC,

DE || BC

∴

[By B.P.T.]

But AD = DB

⇒ AD/DB = 1

⇒ 1 = AE/EC ⇒ AE = EC

Hence, DE bisects AC.**Q.8. Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.****Sol.** Given. A ΔABC in which D and E are mid-points of sides AB and AC respectively.

To Prove: DE || BC

Proof: In ΔABC, AD = DB and AE = EC

∴ DE || BC [By converse of B.P.T.]**Q.9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that ****.Sol.** Given: ABCD is a trapezium in which AB || DC

Construction: Draw EO || DC

Proof: In ΔABD, EO || DC [By construction]

DC || AB [Given]

⇒ EO || AB

From equation (i) and (ii)

**Q.10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that ****. Show that ABCD is a trapezium.****Sol. **Given: A quadrilateral ABCD, whose diagonals intersect at O.

To Prove: ABCD is a trapezium.

Construction: Draw EO || AB

Proof: In ΔABC, OE || AB

∴

From equation (i) and (ii)

⇒ OE || DC [By converse of B.P.T.]

OE || AB and OE || DC ⇒ AB || DC

∴ ABCD is a trapezium.

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