CHAPTER NUMBER — 6

TRIANGLES

KEY CONCEPTS:

We know that two geometric figures are said to be congruent, if they

have the same shape and the same size.

Two geometric figures having the same shape (but not necessarily the

same size) are called similar figure.

Similar Polygons:

l) The lengths of their corresponding sides are proportional.

2) Their corresponding angles are equal.

Two congruent figures are similar , but two similar figures need

not be congruent.

All line segments are similar.

All equilateral triangles are similar.

All squares are similar.

All circles are similar.

If a line is drawn parallel to one side of a triangle to intersect the

other two sides in distinct points, then the other two sides are

divided in the same ratio.

If a line divides any two sides of a triangle in the same ratio, then the

line is parallel to the third side.

If in two triangles, corresponding angles are equal, then their

corresponding sides are in the same ratio and hence the two triangles

are similar (AAA similarity criterion).

If in two triangles, two angles of one triangle are respectively equal to

the two angles of the other triangle, then the two triangles are similar

(AA similarity criterion).

If in two triangles, corresponding sides are in the same ratio and their

corresponding angles are equal then two triangles are similar (SSS

similarity criterion).

If one angle of triangle is equal to one angle of another triangle and the

sides including these angles are in the same ratio (proportional), then

two triangles are similar (SAS similarity criterion).

The ratio of areas of two similar triangles is equal to the square of the

ratio of their corresponding sides.

In a right triangle, the square of the hypotenuse is equal to the sum of

the squares of the other two sides (Pythagoras Theorem).

If in a triangle, square of one side is equal to the sum of the squares of the

other two sides. Then the angle opposite to the first side is a right angle.

If in two right triangles, hypotenuse and one side of one triangle are

proportional to the hypotenuse and one side of the other triangle, then the

two triangles are similar. ( RHS similarity criterion) .

SECTION – A

MULTIPLE CHOICE QUESTIONS

Q1. ABC and BDE are two equilateral triangles such that D is the mid-point

of BC. Ratio of the area of triangles ABC and BDE is:

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

Q2. Sides of two similar triangles are in the ratio of 4:9. Area of these

triangles is in the ratio of:

a) 2:3 b) 4:9 c ) 81:16 d ) 16:81

Q3. In AD = x + 1, DB = x – 1, AE = 2x and EC = x, find the

value of x

a) 3 b) 4 c ) 3.5 d ) 6

Q4. In AD is the bisector of A, meeting the side BC at D. if

AB = 5.6cm, AC = 6cm and DC = 3cm, find BC.

a )4cm b)4.9cm c) 5.8cm d) 4.5cm

Q5. A street light bulb is fixed on a pole 6m above the level of the street. If a

women of height 1.5m casts a shadow of 3m on the ground, then find how

far she is away from the base of the pole.

a )11m b)9m c)10m ( d )13m

FILL IN THE BLANKS

Q6. If the corresponding sides of two similar triangle are in the ratio 5:7, then

their perimeters are in the ratio_______.

Q7. If the corresponding sides of two similar triangle are in the ratio 4:3, then

their areas are in the ratio_______.

Q8. If ABC DEF and A = 470, E = 83

0, C = _______.

Q9. If the ratio of the corresponding sides of two similar triangles is 7:11,

then the ratio of their corresponding altitudes is _________.

Q10. If the corresponding medians of two similar triangle are in the ratio 3:5,

then the ratio of their corresponding sides is _______.

ANSWER THE FOLLOWING

Q11. The dimensions of one rectangle are 4cm and 5cm. The dimensions

of another rectangle are 4cm and 6cm, Are the rectangles similar?

Q12. In ABC, AB = 24cm, BC = 10cm and AC =26cm. Is this triangle a right

triangle ? Give reasons for your answer.

Q13. State whether the following pair of figure is similar or not.

Q14. State whether the following pair of figure is similar or not.

Q15. State whether the following pair of figure is similar or not .

SECTION – B

Q16. The length of a diagonal of a rhombus is I6cm. If its side is 10cm,

then find the length of the other diagonal.

Q17. In the given ABC, P and Q are points on the sides AB and AC

respectively of ABC such that AP = 3.5 cm, QC=6cm. If

PB=4.5cm, find AQ, if PQ ∥ BC .

Q18. In the figure, PQ=24cm, QR = 26cm, PAR = 900,

PA = 6cm and AR = 8cm. Find QPR.

Q19. If the diagonals of a quadrilateral divide each

other proportionally, prove that it is a trapezium.

Q20. In a right ABC, the perpendicular BD on hypotenuse AC is drawn

Prove AC.CD = BC2.

SECTION – C

Q21. A ladder 10 m long reaches a window 8m above the ground. Find

the distance of the foot of the ladder from base of the wall.

Q22. Two poles of heights 6m and 11m stand on a plane ground. If the

distance between the feet of the poles is 12m, find the distance

between their tops.

Q23. A vertical pole of length 6m cast a shadow 4m long on the ground

and at the same time a tower casts a shadow 28m long. Find the

height of the tower.

Q24. If AD and PM are medians of ABC and PQR, respectively

where ABC PQR, Prove that (AB/PQ) =(AD/PM).

Q25. D, E and F are respectively the mid-points of sides AB, BC and CA

of ABC. Find the ratio of the areas of DEF and ABC.

SECTION — D

Q26. Prove that the ratio of the areas of two similar triangles is equal to

the square of the ratio of their corresponding medians.

Q27. 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.

Q28. Prove that the sum of the squares of the sides of a rhombus is equal

to the sum of the squares of its diagonals.

Q29. ABC is a right triangle right angled at B. Let D and E be any points

on AB and BC respectively. Prove that AE2 +CD

2=AC

2+ DE

2.

Q30. 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.

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CHAPTER NUMBER — 6

TRIANGLES

ANSWER KEY

SECTION- A

MULTIPLE CHOICE QUESTIONS

Q1. c) 4:1

Q2. d) 16:81

Q3. a) 3

Q4. c) 5.8cm

Q5. b) 9m

FILL IN THE BLANKS

Q6. 5:7

Q7. 16:9

Q8.

Q9. 7:11

Q10. 3:5

ANSWER THE FOLLOWING

Q11. No, They are not similar.

Q12.

Q13. No, They are not similar.

Q14.

Q15. No, They are not similar.

SECTION B

Q16. Diagonals of rhombus are bisector of each other.

Q17.

= 14/3

Q18.

R,

………………(2)

Q19.

(SAS similarity criteria)

is trapezium

Q20.

Section-C

Q21.

Q22.

Q23. In

⸫

(by AA similarity criteria)

Q24.

In ,

Proved

Q25. Given:

&

=

=

=

SECTION – D

Q26. Given:

But

Similarly we can prove

Q27.

(1)

,

⸫

Q 28. Given:

,

⸫

Q 29. Given:

ii) D lies on AB

iii) E lies on BC

TPT:

Proof: In right angled ,

In right angled ,

Add (1) & (2) +

( + )

Q 30. Given:

ii) AD BC, BE AC ,CF AB

TPT:

Proof : In

In right angled ,

=

Hence proved.

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