Triangles Exercise 6.2

 Hi Friends and Champs,

Before proceeding to the next exercise let us understand following:

Triangle Similarity:

Two triangles will be similar if and only if:

1. their corresponding angles are equal, and

2. their corresponding sides are in the same ratio.

For example, in the below given figure:

In Δ ABC and ΔADE , if both the triangles are similar:
then <ADE = <ABC and <AED = <ACB 

and AD / AB = AE /AC

Thales Truth: The ratio of any two corresponding sides in two equiangular triangles will always be same.   ( Note:Do not confuse in equiangular and equilateral)

From the above facts, we can prove that:

Theorem-6.1 -If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. (For proof, pls see Book and try it first by yourself)

According to the theorem , it is given that DE || BC , then AD / DB = AE / EC

and the reverse is also true 

Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. ( For proof, pls see Book and try it first by yourself)

According to the theorem, if AD / DB =AE / EC then DE || BC.

Depend on these theorems, we will do the next exercise. 

                                                                        Exercise 6.2

 1. In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).

Fig 6.17

Solution:

2. E and F are points on the sides PQ and PR respectively of a Δ PQR. 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

Solution :

                                                            => EF || QR

 3. In Fig. 6.18, if LM || CB and LN || CD, prove that AM/AB=AN/AD  

Fig 6.18

Solution:

 4. In Fig. 6.19, DE || AC and DF || AE. Prove that BF/FE = BE/EC.

Solution:

 5. In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.

Solution:

 6. In Fig. 6.21, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.

Fig 6.21

Solution:

 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. (Recall that you have proved it in Class IX). 

Solution :

 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. (Recall that you have done it in Class IX).

Solution:

 9. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO/BO = CO/DO.

Solution:

10. The diagonals of a quadrilateral ABCD intersect each other at the point O such that 

 AO/BO = CO/DO Show that ABCD is a trapezium

Solution:

Thanks for reading, kindly comment for your suggestions. I will definitely work on that.