Chapter 6. TRIANGLES 

Class 10 Maths Chapter 6: Triangles Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :                                           

 Class 10 Triangles Multiple Choice Questions and Answers

SECTION = A

Q1. If  and  ar() =  ar(), then the value of  is :

(a)              (b)           (c)           (d)   

Solution:  (c)           

  [     Since, DEF  ABC .

   Given ,

       

        ] 

Q2. In given figure , S and T are points on the sides PQ and PR , respectively of ∆PQR , such that PT = 2 cm and TR = 4 cm and ST is parallel to QR ,then the ratio of the  area of ∆PST and ∆PQR    :

   

  (a)              (b)              (c)             (d) 

 Solution:   (b)   

 [ Since STQR  , then    .

 A/Q , 

                 ]

Q3. In figure , DEBC ,then EC is equal to :

                    

(a)  2 cm          (b) 3 cm       (c) 5 cm      (d) 6 cm

Solution: (a)  2 cm     

 [  In   and  we have ,

     

  cm   ]

Q4. Which of the following  given the sides of the triangle make is a right triangle ?

 (a)  3 cm , 8 cm , 6 cm        

(b) 50 cm  , 80 cm , 100 cm

 (c) 25 cm , 24 cm , 7 cm                              

(d)  7 cm , 11 cm , 13 cm  

Solution:  (c) 25 cm , 24 cm , 7 cm                       

 [ Here ,  cm  ,  cm  and   cm

   

   Therefore ,   is a right triangle .       ]

Q5.  ABC and BDE are two equilateral triangles such that D is the mid-point of BC . Ratio of the areas of triangles ABC and BDE is  :

(a)  2 : 1       (b) 1 : 2         (c) 4 : 1        (d) 1 : 4

Solution:   (c) 4 : 1  

 [   Since  ABC and BDE ae two equilitarel triangles , i.e.,  ABC BDE .     

    

 So,         4 : 1   ]     

Q6.  If  and , then RQ is :

 (a) 6 cm       (b)  12 cm      (c) 10 cm     (d) 3 cm

Solution:   (b)  12 cm 

[    Since  , we have

    So,   

  

            ]

Q7.  Let ABC be a triangle such that AB =  cm , AC = 12 cm and BC = 6 cm ,then  is :   

 (a)  120°       (b) 60°      (c)  90°     (d) 45°

Solution:   (c) 90° 

 [ InABC , we have  

 

 

  . 

 So,     ] 

Q8.  In given figure , MNAB , AB = 7.5 cm , AM = 4 cm and MC = 2 cm , then the length of BN is :

  

 (a) 5 cm         (b)  4 cm          (c)  2 cm       (d) 8 cm

Solution:   (a) 5 cm

[ In ∆ABC and MNAB , then    ABCMNC ,

we have      

  

 cm    cm       ]                                                                  

Q9.  In DEW , ABEW . If  and  ,then the value of DB is : 

 

    (a) 12 cm         (b) 24 cm       (c) 8 cm        (d) 4 cm

Solution:  (c) 8 cm

 [      InDEW and ABEW,

We have,   

 

  cm          ]

Q10. DE is drawn parallel to the base BC of a ABC , meeting AB at D and AC at E . If  and CE = 2 cm , then  AE is :

 (a)  5 cm        (b)  4 cm      (c) 6 cm        (d) 7 cm 

Solution:   (c) 6 cm 

 [   In ∆ABC and DEBC,                           

 

We have , 

  . 

 So,         cm       ] 

  Class 10 Triangles  Fill in the blank :

Q1.  All circles are .  [ congruent / similar]

Solution:  Similar .

Q2. All squares are   .  [ similar / congruent ]

Solution:  Similar .

Q3. All    triangles are similar .   [ isosceles / equilateral / acute triangle  ]

Solution:  Equilateral .

Q4. Two polygons of the same number of sides are similar , if (a) their corresponding angles are  and (b) their corresponding sides are  . [congruent / equal / proportional /Similar ]

Solution:  Equal  , Proportional .

Q5.   is an isosceles triangle in which  90° . If AC= 6 cm , then .                   

Solution:    cm

[ In  ,we have

               

    [ ∵  BC = AC ]

  cm     ]     

 Class 10 Triangles 2 Marks Questions and Answers                                                      

SECTION = B

Q1. ABC is an equilateral triangle of side 2a . Find each of its altitudes .

Solution:   Since ABC be an equilateral triangle  AB = BC = AC =  

 We draw  ADBC   then

     

In  , we have     

  

 

 

 

 

   

Q2.  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  and  .

To prove :    .

Proof : In given figure,

In ABC , we have

                                  

  

i. e.  

 In  and  , we have   

 

    [ Given]

     [ A.A rule ]     Proved  .

Q3.  In given figure, if  and  ,  prove that    .

  

Solution: In given figure,

  In  and   we have ,

     

Again,   and  we have ,  

     

  and we have ,    

 

 

 

 

              Proved.

Q4. In given figure , If  ,  prove that .

  

Solution:  In given figure,

In  , we have 

 

 In , we have                          

  

 

  

      Proved . 

 Class 10 Triangles 3 Marks Questions and Answers                                                     

SECTION = C

Q1.  D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C . Prove that  .  

Solution:   Given,  and are points on the sidesand respectively of a triangle  right  angled  at C .

To Proved  :   

Proof : In given figure,

   In  we have,                 

       

In  we have,

  

 In  we have,

  

 In  we have,

                    

    and  we get ,

 

   [ From  and  ]

   Proved .

Q2. In figure , ABC and DBC are two triangles on the same base BC . If AD intersects BC at O , show that     .       

    

Solution:  Given,  and  are two triangles on the same base and  intersects  at O .

To Prove :      

Construction :      We draw   and  .

Proof : In given figure ,

  In  and     We have ,

         

 [ Vertical opposite angle]

     [ Alternative interior angle]

  [A-A-A rule]

So ,       ……………… (i) 

        [ From (i) ]    Proved  .

  Class 10 Triangles 4 Marks Questions and Answers                                             

SECTION = D

1. ABCD is a trapezium with AB∥DC . E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB . Show that    .

                             

Solution:  Given, ABCD is a trapezium with ABDC . E and F are points on non-parallel sides AD  and BC respectively such that EF AB .

To Prove :          

Construction :  Join AC to intersect EF at G .

Proof : In given figure,

ABDC and EFAB and Also EFDC

 In ADC and EGDC        (  EFDC )

So,          ………….. (i)

 In ACB and GFAB  ( GFAB )

So,             ………….. (ii)

From (i) and (ii) , we get

       Proved.  

2. In a triangle, if square of one side is equal to the sum of the squares of the other two sides , then the angle opposite the first side is a right angle . 

Solution:  Given , ABC be a triangle in which   .

To Prove :    

 Construction : We draw a  right angled at Q such that PQ = AB and QR = BC .

 Proof : In figure,

   In  , we have

       [   ]

     [ PQ = AB  and QR = BC] 

Again ,     

 From  and  , we get  

In  and   , we have

    

   

     [ S.S.S congruence]

       [ CPCT]  

        Proved .  

3. In figure, the line segment XY is parallel to side AC of ABC and it divides the triangle into two parts of equal areas . Find the ratio  .    

   

Solution: In given figure,

Since, the line segment XY is parallel to side AB of the triangle ABC .

Therefore,    and     [ corresponding angles]

 In  and  , we have

    [ Given]

 [ Given]

         [ A.A rule]   

 So,           

Given,    

   

  From  and  , we get   

 

 

 

 

 

   

 

   

Q4.  The perpendicular from A on side BC of a ABC intersects BC at D such that   . Prove that  .

Solution:   Given,  be a triangle and  and  intersect at such that . 

To prove :    

Proof : In figure,

Given,  

  

         

 and        

 In   we have ,

 

In   we have ,

 

 

      

    

   

   

  

   

 

  

       Proved. 

 5. In a right triangle , the square of the hypotenuse is equal to the sum of the squares of the other two sides .

Solution:  In figure,

Given , ABC be a right triangle and B = 90° .

To prove :      .

Construction : we draw BDAC .

Proof : In  and  , we have

   [ Common angle]

  

           [ A. A. ]   

So,            

   

  In  and  , we have

    [ Common angle]

  

         [ A. A. ]

 So,          

  

Adding  and  , we get  

 

          Proved .

6. Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides .

Solution:   Given , ABCD be a parallelogram and diagonals AC and BD intersecting at a point O .

To Prove :    .

Proof : In figure,

We  know that  diagonals of a parallelogram bisect each other .

  i.e. ,   and    

Since OB be a median of ABC ,  then

           …………… (i)

Again,   OD be a median of  ADC , then

          …………… (ii)

Adding (i) and (ii) , we get   

 

 

         Proved .   

 Class 10 Triangles 5 Marks Questions and Answers                                                 

SECTION = E

Q1.  Prove that  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 .

Solution:  Given, a triangle ABC in which a line parallel to side BC intersects other two sides AB and AC at D and E respectively .

To prove :        .

Construction : Join BE and CD  and also , draw DMAC and ENAB .

Proof : In figure,

We know that , Area of triangle   × Base × Height 

 

    

 

 

and    

    

   and       

   Since  and  are on the same base DE and between the same parallels BC and DE.

So,        

From  ,   and  , we  have          

    

      Proved.       

Q2. In figure,  O is any point inside a rectangle ABCD . Prove that  .

Solution:  Given,  is any point inside a rectangle ABCD .

To Prove :    .

Construction :  Through O, draw PQBC and P lies on AB and Q lies on DC .

Proof : In given figure,

 Since,  and  .

      

 and     

 Therefore ,  and   are both rectangles .

 In    we have ,

  

 In  we have ,

  

 In  we have ,

  

 In  we have ,

  

  

       [,]

 

    Proved . 

Q3.  In figure , O is a point in the interior of a triangle ABC ,  , and  .  Show that   

  

Solution: Given O is a point in the interior of a triangle ABC ,  ,  and  .

To Prove  :  

Construction :  We join , and  

 Proof :  In figure,

 

  In  we have ,

     

    

 In  we have ,

    

  

 In  we have ,

    

  

 

    

In  we have ,

    

 

In  we have ,

    

  

 In   we have ,

      

   

  

 

From  and  , we have

          Proved.

Q4.  Prove the ratio the areas of two similar triangles is equal to the square of the ratio of their corresponding  sides . [CBSE 2020 standard]

Solution:   Given, ABC and PQR are two triangle such that  .

To Prove  :           

Construction :  We draw AMBC and PNQR .

Proof : In figure,

  In  ABC , we have

                    

In  PQR , we have

 

In ABM and PQN , we have 

   [     ]        

 In   [  A.A  ]

         

In   [  Given  ]

     

                 [ From  and ]

Similarly , we show that    Proved.