polycet maths materials

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8.Similar Triangles

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Similar Triangles
1. Two figures having the same shape but not necessarily the same size are called
similar figures
2. All congruent figures are similar but the converse is not true.
3. Two polygons having the same number of sides are similar,
i. Their corresponding angles are equal and
ii. Their corresponding sides are proportional (i.e., in the same ratios)
4. 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.
5. If a line divides any two sides of a triangle in the same ratio, then the line is
parallel to the third side of the triangle.
6. The internal bisector of an angle of a triangle divides the opposite side internally
in the ratio of the sides containing the angle.
7. If a line through one vertex of a triangle divides the opposite side in the ratio of
other two sides, then the line bisects the angle at the vertex.
8. The external bisector of an angle of a triangle divides the opposite side externally
in the ratio of the sides containing the angle.
9. The line drawn from the midpoint of one side of a triangle is parallel of another
side bisects the third side.
10. The line joining the mid points of two sides of a triangle is parallel to the third
side.
11. The diagonals of a trapezium divide each other proportionally,
12. If the diagonals of a quadrilateral divide each other proportionally then it is a
trapezium
13. Any line parallel to the parallel sides of a trapezium divides the non-parallel
sides proportionally.
14. If three or more parallel lines are intersected by two transversals, then the
intercepts made by them on the transversals are proportional.
15. AAA Similarity criterion: If in two triangles, corresponding angles are equal,
then the triangles are similar.

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16. AA 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.
17. SSS Similarity criterion: If in two triangles, corresponding sides are in the same
ratio, then the two triangles are similar.
18. If one angle of triangles is equal to one angle of another triangle and the sides
including these angles are in the same ratio, then the triangles are similar.
19. If two triangles are equiangular, then
i. The ratio of the corresponding sides is same as the ratio of corresponding
medians.
ii. The ratio of the corresponding sides is same as the ratio of the
corresponding angle bisector segments.
iii. The ratio of the corresponding sides is same as the ratio of the
corresponding altitudes.
20. If one angle of a triangle is equal to one angle of another triangle and the
bisectors of these equal angles divide the opposite side in the same ratio, then
the triangles are similar.
21. If two sides and a median bisecting one of these sides of a triangle are
respectively proportional to the two sides and the corresponding median of
another triangle, then the triangles are similar.
22. If two sides and a median bisecting, the third side of a triangle are respectively
proportional to the two sides and the corresponding median of another triangle,
then the triangles are similar.
23. The ratio of the areas of two similar triangles is equal to the ratio of
i. The squares of any two corresponding sides.
ii. The squares of the corresponding altitudes.
iii. The squares of the corresponding medians.
iv. The squares of the corresponding angle bisector segments.
24. If the areas of two similar triangles are equal, then the triangles are congruent
i.e., equal and similar triangles are congruent.
25. If a perpendicular is drawn from the vertex of the right angled triangle to the
hypotenuse, then the triangles on both sides of the perpendicular are similar to
the whole triangle and also to each other.

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26. Pythagoras Theorem: In a right angled triangle, the square of the hypotenuse is
equal to the sum of the squares of the other two sides.
27. Converse of 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 first side
is aright angle.
28. In any triangle, the sum of the squares of any two sides is equal to twice the
square of half of the third side together with the twice of the square of the
median which bisects the third side.
29. Three times the sum of the squares of the sides of a triangle is equal to four times
the sum of the squares of the medians of the triangle.
30. Three times the square of nay side of an equilateral triangle is equal to four times
the square of the altitude.

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Multiple Choice Questions
1. In triangle ST is a line such that and also then
is
(a)Right angled triangle (b)Isosceles triangle
(c)Both (a) & (b) (d)None
2. In the figure, and then which of the following is true
(a) (b) (c) (d)None
3. In and and given the
(a) (b)5cm (c)2.1cm (d)5.6m
4. In the given figure
Then the value of
(a) (b)8 (c)7 (d)6
5. If the diagonals of a quadrilateral ABCD intersect each other at point “O” such
that . Then ABCD is a
(a) Square (b)Parallelogram
(c)Rhombus (d)Trapezium
6. A person 1.65m tall casts 1.8m shadow. At the same instance, a lamp post casts a
shadow of 5.4m. Then the height of the lamp post

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(a) (b)4.95m (c)1.89m (d)None
7. A man sees the top of a tower in a mirror which is at a distance of 87.6m from the
tower. The mirror is on the ground facing upwards. The man is 0.4m away from
the mirror and his height is 1.5m then the height of the tower
(a)300m (b)325m (c)328m (d)326m
8. If the perimeters of two similar triangles are 30cm and 20cm respectively. If one
side of the first triangle is 12cm, the corresponding side of the second triangle.
(a)6c (b)7cm (c)8cm (d)9cm
9. A girl of height 90cm is walking away from the base of a lamp post at a speed of
. If the lamp post is 3.6m above the ground, then the length of her
shadow after 4 seconds.
(a)1.6 (b)1.5m (c)1.4m (d)1.7m
10. AX and DY are altitudes of two similar triangles and then which of
the following is correct
(a)AB:BC = AX:DY (b)AX:DY = AB:DE
(c)AC:AX = DY:AB (d)None
11. A flag pole 4m tall casts a 6m shadow. The same time, a near by building casts a
shadow of 24m. Then the height of the building is
(a)14m (b)15m (c)16m (d)17m
12. CD and GH are respectively the bisectors of and such that D and H
lie on sides AB and FE of and respectively. If . Then
which of the following is correct.
(a) (b)
(c) (d)All of the above
13. and their areas are respectively and . If
EF=15.4cm, then BC=
(a)11.2cm (b)13.2cm (c)14.1cm (d)None
14. If the diagonals of a trapezium ABCD with intersect each other at the
point “O” If the ratio of the triangles . Then which of the
following is true
(a)AB=CD (b)AB=2CD (c)AB=3CD (d)None

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15. If three equilateral triangles are drawn on the three sides of right angled triangle
and the areas of triangles drawn on the two sides are and
respectively, then the area of the triangle drawn on the hypotenuse is
(a)3025 (b)70 (c)1125 (d)both (a)&(b)
16. are mid points of sides of then the ratio of areas of
and is
(a)1:1 (b)1:2 (c)4:1 (d)None
17. In and XY divides the triangle into two parts of equal area. Then
(a) (b) +1 (c) -1 (d)None
18. The areas of two similar triangles are 81 and 49 respectively. If the
altitude of the bigger triangle is 4.5cm. Then the corresponding altitude for the
same triangle.
(a)4.5cm (b)2.5cm (c)1.5cm (d)3.5cm
19. A ladder 25m long reaches a window of building 20m above the ground. Then
the distance of the foot of the ladder from the building.
(a)15m (b)25m (c)20m (d)45m
20. The hypotenuse of aright angled triangle is 6m more than twice of the shortest
side. If the third side is 2m less than the hypotenuse, the sides of a triangle.
(a)5m, 12m, 13m (b)3m, 4m, 5m (c)10m, 24m, 26m (d)None
21. is a right angled triangle, right angle at P and M is appoint on QR such that
. Then which of the following is correct.
(a) (b) (c)both (a) & (b) (d)None
22. is an isosceles right angled triangle right angle at C, then
(a) (b) (c) (d)both (a) & (b)
23. Two poles of height 6m and 11m stand on a plane ground. If the distance
between the feet of the poles is 12m then the distance between their tops is
(a)13m (b)12m (c)11m (d)10m
24. In an equilateral triangle D is a point on the side BC such that
Then which of the following is true.
(a) (b) (c) (d)

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25. 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
26. The areas of two similar triangles and are 144 and 81
respectively. If the longest side of larger be 36cm, then the longest side of
the smaller triangle
(a)20cm (b)26cm (c)27cm (d)30cm
27. and are two equilateral triangles such that D is the midpoint of BC.
The ratio of the areas of triangles and is
(a)2:1 (b)1:2 (c)4:1 (d)1:4
28. Two isosceles triangles having equal angles and their areas are in the
ration16:25. The ratio of their corresponding heights is
(a)4:5 (b)5:4 (c)3:2 (d)5:7
29. If and are similar such that and BC=8cm, then EF=
(a)16cm (b)12cm (c)8cm (d)4cm
30. If is such that AB= 3cm, BC= 2cm and CA= 2.5cm. If and
EF=4cm, then perimeter of is
(a)7.5cm (b)15cm (c)22.5cm (d)30cm
31. In a , is a line XY parallel to BC cuts AB at X and AC at Y. If BY bisects
then
(a)BC=CY (b)BC=BY (c) (d)
32. In a D and E are points on side AB and AC respectively such that
and If EA=3.3cm, then AC=
(a)1.1cm (b)4cm (c)4.4cm (d)5.5cm
33. In triangles and , and
then
(a) (b) (c) (d)
34. If and are similar such that , then
(a) (b) (c) (d)
35. In a , AB=5cm and AC=12cm, If then AD=
(a) (b) cm (c) cm (d) cm
36. In an equilateral triangle ABC, if then
(a) (b) (c) (d)

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37. If is an equilateral triangle such that that then
(a) (b) (c) (d)4
38. In a perpendicular AD from A on BC meets BC at D. If BD=8cm, DC=2cm
and AD=4cm then the is
(a) Isosceles (b)Equilateral
(c)Scalene (d)Right angled
39. In a AD is the bisector of if AB=6cm, AC= 5cm and BD=3cm, then
DC=
(a)11.3cm (b)2.5cm (c)3.5cm (d)none
40. In a AD is the bisector of if AB=8cm, DC= 3cm and BD=6cm, then
AC=
(a)4cm (b)6cm (c)3cm (d)8cm
41. is an isosceles triangle and D is a point on BC such that , then
(a) (b)
(c) (d)
42. If E is a point on side CA of an equilateral triangle ABC such that then
(a) (b) (c) (d)
43. and their areas are respectively and . If
, then the measure of EF is
(a)2.8cm (b)4.2cm (c) cm (d) cm
44. The length of the hypotenuse of an isosceles right angled triangle whose one side
is cm is
(a)12cm (b)8cm (c) cm (d) cm
45. A man goes 24m due west and then 7m due to north. How far is he from the
starting point.
(a)31m (b)17m (c) (d)
46. In an equilateral triangle ABC, if then
(a) (b) (c) (d)
47. If and are similar such that AB=9.1cm and DE=6.5cm. If the
perimeter of is 25cm, then the perimeter of is
(a)36cm (b)30cm (c) (d)

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48. In an isosceles triangle if and , then
(a) (b) (c) (d)
49. Is an isosceles triangle in which . If AC=6cm, then AB=
(a) cm (b) (c) cm (d) cm
50. In an isosceles triangle , if and BC=14cm, then the
measure of altitude from A on BC is
(a) cm (b) (c) cm (d) cm
Answers
1 2 3 4 5 6 7 8 9 10
b a c a b b c c a b
11 12 13 14 15 16 17 18 19 20
c d a c b c c d a c
21 22 23 24 25 26 27 28 29 30
b c a d d c c a d b
31 32 33 34 35 36 37 38 39 40
a c b a b c c d b a
41 42 43 44 45 46 47 48 49 50
a c a b c b d c a d