6. TRIANGLES (SCERT)

SEBA Class 10 Maths Chapter 6. TRIANGLES

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

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

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]

[ 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:

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.

Class 10 Maths Chapter 6. TRIANGLES Exercise 6.4 Solutions

1.1. Let and their areas be, respectively , 64 and 121 . If , find BC .

Solution: In figure ,

Since, , we have

2. Diagonals of a trapezium ABCD with intersect each other at the point O . If , find the ratio of the areas of triangles AOB and COD .

Solution: In figure,

In and , we have

[Vertically Opposite angle]

[Alternative interior angle]

[AAA similarity criterion ]

3. In Fig. 6.44, ABC and DBC are two triangle on the same base BC . If AD intersects BC at O , show that

Solution: In given figure,

Solution: Given, ABC and DBC are two triangle on the same base BC . AD intersects BC at O .

Then we show that :

Construction: We draw and .

Proof : In given figure,

In and , we have

[Vertically opposite angle]

[Third angle]

[AAA similarity criterion]

From (i) and (ii) , we get

Proved.

4. If the areas of two similar triangles are equal , prove that they are congruent .

Solution: let ABC and DEF are two triangles and . To prove: .

Proof: In given figure,

Since, , We have

But

Again,

and

In and , we have

[Given]

[SSS rule] Proved

5. D , E and F are respectively the mid-points of sides AB , BC and CA of .Find the ratio of the areas of and .

Solution: Given, D , E and F are respectively the mid-points of sides AB , BC and CA of .

Using mid-point theorem :

Since, E and F are the mid-point of the side BC and AC of the triangle ABC respectively.

and

Again, and

and

Now,

we find BEFD , DECF and ADEF are the three parallelogram .

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

In and , we have

[ Parallelogram BEFD ]

[ Parallelogram DECF ]

[ Parallelogram ADEF ]

[AAA similarity criterion]

6. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians .

Solution: let ABC and PQR are the two triangle and .

To prove that :

Proof: In figure,

Since, D and M are the mid-point of the triangle ABC and PQR respectively .

and

Given, , we have

In and , we have

[Given ]

[ from (i) ]

[SAS similarity criterion]

Again,

[ From (ii) ]

Proved .

7. Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals .

Solution: let BDF and CDE are two equilateral triangle on the diagonal BD and the side CD of the square ABCD respectively .

To prove :

Proof: In figure ,

Since, ABCD is a square .

So, AB = BC = CD = AD

In , we have

[ is a square]

Since, BDF and CDE are two equilateral triangle .

So, [AAA or SSS similarity criterion]

Proved .

Tick the correct answer and justify :

8. ABC and BDE are two equilateral triangles such that D is the mid-point of BC . Ratios 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

[ In given figure :

Since, ABC and BDE are two equilateral triangles .

So,

[ ]

]

9. Sides of two similar triangles are in the ratio 4 : 9 . Areas of these triangles are in the ratio :

(A) 2 : 3 (B) 4 : 9 (C) 81 : 16 (D) 16 : 81

Solution: (D) 16 : 81

[ let ABC and PQR are two triangles . Here,

]

Class 10 Maths Chapter 6. TRIANGLES Exercise 6.5 Solutions

1. Sides of triangles are given below . Determine which of them are right triangles . In case of a right triangle , write the length of its hypotenuse .

(i) 7 cm , 24 cm , 25 cm (ii) 3 cm , 8 cm , 6 cm (iii) 50 cm , 80 cm , 100 cm (iv) 13 cm , 12 cm , 5 cm

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

Let ABC is a any triangle whose sides are AB = 7 cm , BC = 24 cm and AC = 25 cm

In , we have

Therefore, ABC is a right triangle at B . The hypotenuse = 25 cm .

(ii) 3 cm , 8 cm , 6 cm

Let ABC is a any triangle whose sides are AB = 3 cm , BC = 6 cm and AC = 8 cm

In , we have

Therefore, ABC is not a right triangle .

(iii) 50 cm , 80 cm , 100 cm

Let ABC is a any triangle whose sides are AB = 50 cm , BC = 80 cm and AC = 100 cm

In , we have

Therefore, ABC is not a right triangle .

(iv) 13 cm , 12 cm , 5 cm

Let ABC is a any triangle whose sides are AB = 5 cm , BC = 12 cm and AC = 13 cm

In , we have

Therefore, ABC is a right triangle at B . The hypotenuse = 13 cm .

2. PQR is a triangle right angled at P and M is a point on QR such that . Show that .

Solution: Given, PQR is a triangle right angled at P and M is a point on QR such that .

Then we show that :

Proof: In figure,

Since PQR is a triangle right angle at P and .

So,

Proved.

3. In Fig. 6.53 , ABD is a triangle right angled at A and . Show that

(i)

(ii)

(iii)

Solution: Given, ABD is a triangle right angled at A and .

Then we show that :

(i)

(ii)

(iii)

Proof: In given figure,

(i) Since, and

So,

(ii) Since, and

So,

(iii) Since, and

So,

Proved.

4. ABC is an isosceles triangle right angled at C . Prove that .

Solution: In figure,

Since, ABC is an isosceles triangle right angled at C .

So, AC = BC

Using Pythagoras theorem ,

Proved .

5. ABC is an isosceles triangle with . If , prove that ABC is a right triangle .

Solution: In figure,

Since, ABC is an isosceles triangle .

So, AC = BC

and

Therefore, ABC is an isosceles triangle right angled at C .

6. ABC is an equilateral triangle of side . Find each of its altitudes .

Solution: In figure,

Since, ABC is an equilateral triangle .

We draw

So,

In , we have

Therefore, the altitude is .

7. Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals .

Solution: We know that the diagonals of a rhombus bisect each other at right angles .

So, and

In figure ,

In , we have

For BOC triangle :

For COD triangle :

For AOD triangle :

Adding (i) , (ii) , (iii) and (iv) , we get

Proved.

8.In Fig.6.54, O is a point in the interior of a triangle ABC , and . Show that

(i) .

(ii) .

Solution: Given, O is a point in the interior of a triangle ABC , and . Show that (i) .

(ii) .

Proof: In given figure,

(i) Using Pythagoras theorem,

In , we have

Adding (1) , (2) and (3) , we get

(ii) Using Pythagoras theorem,

In , we have

Adding (5) , (6) and (7) , we get

From (4) and (8) , we get

Proved.

9. A ladder 10m long reaches a window 8m above the ground . Find the distance of the foot of the ladder from base of the wall .

Solution: In given figure,

Here, AB = 8 m , AC = 10 m and BC = ?

In , we have

Therefore, the distance of the foot of the ladder from base of the wall is 6 cm .

10. A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attacked to the other end . How far from the base of the pole should the stake be driven so that the wire will be taut ?

Solution: In given figure,

Here, AB = 18 m , AC = 24 m and BC = ?

In , we have

Therefore, the distance between the base of the pole and the stake is .

11. An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour . At the same time , another aeroplane leaves the same airport and flies due west at a speed of 1200km per hour . How far apart will be the two planes after hours ?

Solution: In given figure,

We know that , The distance = Speed × time

Here,

And AB = the distance between the two planes .

In , we have

Therefore, the distance between the two planes is .

12. Two poles of heights 6 m and 11 m stand on a plane ground . If the distance between the feet of the poles is 12 m , find the distance between their tops .

Solution: In given figure,

Here, AD = 6 m , DE = 12 m and CE = 11 m

So, AD = BE = 6 m , AB = DE = 12 m

In , we have

Therefore, the distance between their tops is 13 m

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

Solution: Given, D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C .

To prove : .

Proof : In figure,

In , we have

Adding (ii) and (iii) , we get

[from (i) and (iv) ]

Proved.

14. The perpendicular from A on side BC of a intersects BC at D such that (SEE Fig. 6.55) . Prove that

Solution: Given, the perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB=3CD . To prove : 2AB²=2AC²+BC² .

Proof: In figure,

Since,

And

In , we have

Proved .

15. In an equilateral triangle ABC , D is a point on side BC such that . Prove that .

Solution: Given, ABC is an equilateral triangle and D is a point on side BC such that .

To prove: .

Construction: We draw .

Proof: In figure,

Since, ABC is an equilateral triangle .

i.e., AB = BC = AC

Again,

So,

and

In , we have

Proved.

16. In an equilateral triangle , prove that three times the square of one side is equal to four times the square of one of its altitudes .

Solution: let ABC is an equilateral triangle and .

To prove : .

Proof: In figure,

Since, ABC is an equilateral triangle .

i.e., AB = BC = AC

In and , we have

[Common side]

[given]

[RHS rule]

[CPCT]

So,

In , we have

Proved.

17. Tick the correct answer and justify : In and . The angle B is :

(A) 120° (B) 60° (C) 90° (D) 45°

Solution: (C) 90°

[ In , We have

i.e.,

So, ABC is a right angled triangle at B . i.e., ]

**Class 10 Maths Chapter 6. TRIANGLES Exercise 6.6 (Optional)* Solutions**

1. In Fig. 6.56, PS is the bisector of of ∆ PQR. Prove that ⋅

2. In Fig. 6.57, D is a point on hypotenuse AC of ∆ ABC, such that , and . Prove that : (i) (ii)

3. In Fig. 6.58, ABC is a triangle in which and AD⊥CB produced. Prove that .

4. In Fig. 6.59, ABC is a triangle in which and AD ⊥ BC. Prove that .

5. In Fig. 6.60, AD is a median of a triangle ABC and AM ⊥ BC. Prove that :
(i)

(ii)
(iii)

6. Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
7. In Fig. 6.61, two chords AB and CD intersect each other at the point P. Prove that :
(i) (ii)

8. In Fig. 6.62, two chords AB and CD of a circle intersect each other at the point P(when produced) outside the circle. Prove that (i) (ii)

9. In Fig. 6.63, D is a point on side BC of ∆ ABC such that ⋅ Prove that AD is the bisector of .

10. Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, how much string does she have out (see Fig. 6.64)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?