6 . TRIANGLE (NCERT)
NCERT
6. TRIANGLES
EXERCISE 6.1
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 : PHOTO
3. State whether the following quadrilaterals are similar or not :
Photo 6.8
EXERCISE 6.2
1. In Fig. 6.17, (i) and (ii) , 
. Find 
in (i) and 
in (ii) .
Solution: (i) Here, 
In 
ABC and DE
BC , we have
(ii) Here, 
In ABC and DEBC , 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
and 
So , 
(ii) Here, and
and 
So ,
(iii) Here, and
and 
and 
So , 
3. In Fig. 6.18, if 
and 
, prove that 
.
Solution : 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 
and , we have
In 
and, we have
From and 
, we get
Proved .
5. In Fig. 6.20, 
and 
. Show that .
Solution: Given , and 
. We show that .
Proof : 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 : Proof : 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 : Since S is a mid-point of PQ , then 
In and 
, we have
and 
, we get
Thus, T is a mid-point of PR .
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 : Since S and T are the mid-point of the side PQ and PR of the triangle PQR respectively .
and 
(i) and (ii) 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 : Given, then and 
.
In 
and ∥PO
, 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 and .
But
and we get , 
So, 
then
∴ ABCD is a trapezium .
EXERCISE 6.3
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 :
2. In Fig. 6.35 , 
and 
. Find 
and 
.
Solution :
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 : In 
and 
, we have
[ Vertically opposite angle]
[ Alternative interior angle]
[ Alternative interior angle]
[ A.A.A. rule ]
Proved .
4. In Fig. 6.36, 
and 
. Show that 
.
5. S and T are points on sides PR and QR of such that 
. Show that
.
Solution :
6. In Fig. 6.37 , if 
, show that 
.
Solution : Given, . Then we show that 
.
Proof : Since,
and 
and 
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)
Solution: (i) In 
and 
, we have
[ Vertically opposite angle]
[Third angle]
[A.A.A. rule]
(ii) In 
and 
, we have
[Common angle]
[Third angle]
[A.A.A. Rule]
(iii) In and , we have
[Common angle]
[Third angle]
(iv) In 
and 
, we have
[Common angle]
[Third angle]
[A.A.A. 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 . Then we prove : ABE 
CFB .
Proof : In ABE and CFB , we have
[ Opposite angle of the parallelogram ]
[ Alternative interior angle]
ABE CFB [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 : (i) In and 
,we have
[Common angle]
[Third angle]
[A.A.A. rule]
(ii) Since , [A.A.A. rule]
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 : (i)
(ii)
(iii)
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 AB = AC and AD
BC and EFAC . Then we prove that : ABD
ECF .
Proof : In ABC , we have
i. e. 
In ABD and ECF , we have
[ Given]
ABDECF [ A.A rule ] Proved .
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 :
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 :
16. If AD and PM are medians of triangle ABC and PQR , respectively where , prove that 
.
Solution:
EXERCISE 6.4
1. Let 
and their areas be, respectively , 64 
and 121 
. If 
, find BC .
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 .
3. In Fig. 6.44, ABC and DBC are two triangle on the same base BC . If AD intersects BC at O , show that 
.
4. If the areas of two similar triangles are equal , prove that they are congruent .
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 .
6. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians .
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 .
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
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
EXERCISE 6.5
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
2. PQR is a triangle right angled at P and M is a point on QR such that 
. Show that 
.
3. In Fig. 6.53 , ABD is a triangle right angled at A and 
. Show that
(i) 
(ii) 
(iii) 
4. ABC is an isosceles triangle right angled at C . Prove that 
.
5. ABC is an isosceles triangle with 
. If , prove that ABC is a right triangle .
6. ABC is an equilateral triangle of side 
. Find each of its altitudes .
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 .
8.In Fig.6.54, O is a point in the interior of a triangle ABC , 
and 
. Show that
(i) 
.
(ii) 
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 .
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 ?
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 ?
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 .
13. D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C . Prove that 
.
14. The perpendicular from A on side BC of a 
intersects BC at D such that 
(SEE Fig. 6.55) . Prove that 
15. In an equilateral triangle ABC , D is a point on side BC such that 
. Prove that 
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 .
17. Tick the correct answer and justify : In 
and 
. The angle B is :
(A) 120° (B) 60° (C) 90° (D) 45°
EXERCISE 6.6 (Optional)*
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) ∆PAC~∆PDB
(ii) PA . PB = PC . PD
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?
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