Chapter 6. TRIANGLES    

Class 10 Maths Chapter 6. TRIANGLES Exercise 6.1 Solutions

1. Fill in the blanks using the correct word given in brackets :

(i) All circles are  . (congruent , similar)

(ii) All squares are  . (similar , congruent)

(iii) All  triangles are similar . (isosceles , equilateral)

(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are  and (b) their corresponding sides are  . (equal , proportional)

Solution :  (i) similar    

(ii)  similar

(iii)  equilateral

(iv)  equal ,  proportional  .

2. Given two different examples of pair of  :

  (i) similar figures        (ii) non-similar figures .

Solution :  (i) Similar figures :

(ii) Non-similar figure :

3. State whether the following  quadrilaterals are similar or not :

 Photo 6.8

Class 10 Maths Chapter 6. TRIANGLES Exercise 6.2 Solutions

1. In Fig. 6.17, (i) and (ii),  . Find in (i) and  in (ii) .

  

Solution:  (i) Here,

In figure,

In  and  , we have

(ii)  Here,

In figure,

 In∆  and  , we have

      

2. E and F are points on the sides PQ and PR respectively of a   . For each of the following cases, state whether   :

(i)   and

(ii) , and

(iii) and

Solution :  (i) Here, and

In given figure,

 

and  

So,  

(ii) Here, ,and

In given figure,

We have,

  

and    

So ,  

(iii)  Here, ,and 

In given figure,

 

Now,   

and  

So,

3. In Fig. 6.18, if   and, prove that

Solution:  In given figure ,

  In  and   we have ,

Again,  and  we have ,

 

 and we have ,  

  Proved .

4. In Fig. 6.19,  and, prove that

   

Solution:  Given,  and.

To prove that :   

Proof : In given figure,

In  and  , we have

In  and, we have

From  and  , we get   

    Proved .

5. In Fig. 6.20,   and. Show that   .

Solution:  Given,  and . Then we show that   .

Proof : In given figure,

In  and  , we have

In  and , we have

From  and  , we get

  Proved .

6. In Fig. 6.21 , A , B and C are points on OP , OQ and OR respectively such that  and  . Show that  .

Solution: Given,  A , B and C are points on OP , OQ and OR respectively such that  and  . Then we show that  .

Proof: In figure ,

   

  In  and  , we have

 

In  and , we have

 

From  and  , We get

 

  Proved .

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: Given, PQR is a triangle whose   and S is a mid-point of the side PQ .

To prove  : T is a mid-point of PR .

Proof : In figure, 

 Since S is a mid-point of PQ , then

      

In  and  , we have

   

 and  , we get

     

Thus, T is a mid-point of PR . Proved .

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: Given, PQR is a triangle such that S and T are the mid-point of the side PQ and PR respectively .

To prove  :   .

Proof : In figure ,

Since S and T are the mid-point of the side PQ and PR  of the triangle PQR respectively .

      

and  

 and  we get

         

Thus,         Proved .   

9. ABCD is a trapezium in which   and its diagonal intersect each other at the point O . Show that  .

Solution:  Given, ABCD is a trapezium in which   and its diagonal intersect each other at the point O . Then we show that :  .

Construction : We join OP such that  .

Proof : In given figure,

Given,  then  and  .

In  and  , we have

    

 and  , we have

    

 and  we get,

   

    Proved .

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

Solution: Given, The diagonals of a quadrilateral ABCD intersect each other at the point O such that .Then we show that ABCD is a trapezium .

Construction : We join OP such that  .

Proof: In given figure,

 

In  and  .

 

But  

 and  we get, 

So,  then

  ABCD is a trapezium . 

                              Proved

Class 10 Maths Chapter 6. TRIANGLES Exercise 6.3 Solutions

1. State which pairs of triangle in Fig. 6.34 are similar . Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :

Solution:  (i)  In given figure,

  In  and  , we have

 

    [AAA]

Yes , Angle-angle-angle (AAA similarity criterion) ,

(ii) In given figure,

In and , we have

  ;

  and  

 

 

Yes , side-side-side (similarity criterion) ,

(iii) In given figure,

 In  and , we have

     ;

  and

   

No ,  and are not similar .

(iv) In given figure,

 In  and , we have

 

  and

    

    [SAS]

Yes , side-angle-side (similarity criterion) , [SAS]

(v) In given figure,

In  and , we have

  

No ,   and are not similar .

(vi) In given figure,

Here,

      

  

 

 In  and  , we have

 

   

  [AAA]                                                                         

Yes , angle-angle-angle (similarity criterion) ,   [AAA]

2. In Fig. 6.35 , and . Find and  .

 

Solution:   Given,  and .

In given figure :

Since, BD is a straight line .

 

In , we have

 

Again,   

So,

 

Therefore, ,  and

3. Diagonals AC and BD of a trapezium ABCD with   intersect each other at the point O . Using a similarity criterion for two triangles , show that   .

Solution:  Given, Diagonals AC and BD of a trapezium ABCD with   intersect each other at the point O . Then we show that  .

Proof : Given figure ,

In and  , we have

  [ Vertically opposite angle]

  [ Alternative interior angle]

  [ Alternative interior angle]

  [ AAA  similarity criterion]

 

      Proved .

4. In Fig. 6.36,    and . Show that  .

Solution: In given figure,

  Since,

So, 

PQR is an isosceles triangle .

Again,   

    [from (i) ]

In  and  , we have

     [Common angle]

   [ given]

 [SAS]

5. S and T are points on sides PR and QR of  such that  . Show that .

Solution:  Given, S and T are points on sides PR and QR of  such that  .

Then we show that  .

Proof : In  and  , we have

 [Given]

  [ Common angle]

  [Third angle ]

 [ AAA similarity criterion]

6. In Fig. 6.37 , if  , show that . Show that .

Solution: Given,  . Then we show that  .

Proof :  Since, , we have

   

  

     

and 

[SAS]      Proved

7. In Fig. 6.38, altitudes AD and CE of   intersect each other at the point P .

Show that : (i)      (ii)    (iii)    (iv)

Solution: Given, altitudes AD and CE of    intersect each other at the point P . Then we show that : (i)      (ii)    (iii)    (iv)

Proof: In given figure,

  (i) In  and  , we have

 

 [ Vertically opposite angle]

  [Third angle]

    [AAA rule]

Proof:  (ii) In  and  , we have

 

  [Common angle]

   [Third angle]

   [AAARule]

Proof: (iii)  In  and  , we have

 

  [Common angle]

   [Third angle]

   [AAA]

Proof: (iv) In  and  , we have

 

   [Common angle]

   [Third angle]

      [AAA rule]   

8. E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F . Show that   .

Solution: Given, E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F .

To prove :   .

Proof: In given figure,

In   and  , we have

  [ Opposite angle of the parallelogram ]

  [ Alternative interior angle]

        [A.A rule]        proved.

9. In Fig. 6.39, ABC and AMP are two right triangles , right angled at B and M respectively . Prove that :  (i)    (ii) 

         

Solution: Given , ABC and AMP are two right triangles , right angled at B and M respectively .

 Then we prove that :   (i)     (ii)

Proof :In given figure,

  (i)  In  and  ,we have

  [Common angle]

  [Third angle]

   [AAA rule]

(ii) Since,     [AAA rule ]

  

   Proved.

10. CD and GH are respectively the bisectors of  and  such that D and H lie on sides AB and FE of  and  respectively . If  , show that :

(i) 

(ii)

(iii)

Solution: Given, CD and GH are respectively the bisectors of  and  such that D and H lie on sides AB and FE of  and  respectively and  . Then we show that :

(i)  

(ii)

(iii)

Proof : In given figure,

  Since, CD and GH are the bisectors of  and  respectively .

and

(i)      In and we have

 [  ]

 [ ]

 [AA]

  

           Proved.

(ii) Proof:  In  and  , we have

  [   ]

 [ ]

  [Third angle]

  [AAA similarity criterion ]

 (iii) Proof:  In  and  , we have

  [  ]

 [ ]

   [Third angle ]

  [AAA Similarity criterion]

11. In Fig. 6.40 , E is a point on side CB produced of an isosceles triangle ABC with  . If  and , prove that  .

Solution:  Given, E is a point on side CB produced of an isosceles triangle ABC with ,  and .

To prove that  .

Proof : In given figure,

In , we have

         AB = AC

  

i. e.  

 In  and  , we have   

  

     [ Given]

  [Third angle]

   [ AAA rule ]

12. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of   (see Fig. 6.41) . Show that   .

Solution : Given , Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of   . Then we show that  .

13. D is a point on the sides BC of a triangle ABC such that  . Show that  .

Solution: Given, D is a point on the sides BC of a triangle ABC such that  . Then we show that  .

Proof: In given figure,

  In  and  , we have

 

    

 

  [A.A.A.]

    

      

Proved.

14. Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR . Show that   .

Solution: Given, Sides AB and AC and median AD of a triangle ABC are respectively proportional to sides PQ and PR and median PM of another triangle PQR .

Then we have to show that ∆ABC  ~ ∆PQR   .

Construction : We draw  and  .

Proof : Since ,  and  , then

           and          [Using mid-point theorem]

  

Also , X and Y is the mid-point of the sides AB and QR , then

     and 

We have ,  

     

        

In  and    , We have

    

   [SSS]

     [CPCT] ………………… (i)

Similarly, we show that   …………………. (ii)

Adding (i) and (ii) , we get 

 

In  and  , we have,

   

    

  [SAS]    Proved .

15. A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long . Find the height of the tower .

Solution : let  and are the two triangle .

In given figure,

Here, AB = 6 m , BC = 4 m , PQ = ?  and QR = 28 cm

In and  , we have

   [Same inclination]

  [Third angle]

     [AAA ]

    

Therefore, the height of the tower is 42 m .

16. If AD and PM are medians of triangle ABC and PQR , respectively where  , prove that 

Solution: Given,  AD and PM are medians of triangle ABC and PQR respectively and  .

To prove :     

Proof :  In figure,

Since , D and M are mid-point of the sides BC and QR .

So,   and

Given,

Then,  

     

   

In  and , we have

  [  ]

         

    [S.A.S.]

       

    Proved.