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1 HPTR 4 SIMILR TRINGLS KY POINTS. Similar Triangles : Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional.. riteria for Similarity : in and F (i) Similarity : ~ F when =, = and = F (ii) SS Similarity : ~ F when and F (iii) SSS Similarity : ~ F,. F F 3. The proof of the following theorems can be asked in the examination : (i) (ii) (iii) asic Proportionality Theorem : If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 4 X Maths
2 (iv) onverse of Pythagoras Theorem : In a triangle, if the 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. MULTIPL HOI USTIONS. ~ F. If = and = 3cm then F is equal to. (a).5 cm (b) 3 cm (c) 6 cm (d) 9 cm. In W, W if = 4 cm, = cm and W = 4 cm then the value of = 3. c (a) 4 cm (b) 8 cm (c) cm (d) 6 cm O a b In the figure the value of cd = (a) ae (b) af (c) bf (d) be f d e O F 4. If in, = 6 cm, = cm and 6 3 of is cm then the measure (a) 30 (b) 45 (c) 60 (d) 90 X Maths 5
3 5. The area of two isosceles triangles are in the ratio 6 : 5. The ratio of their corresponding heights is (a) 5 : 4 (b) 3 : (c) 4 : 5 (d) 5 : 7 6. In the figure, is similar to 53 6 cm 36 cm 53 4 cm (a) (b) (c) (d) 7. M ~ M. lso ar (M) = ar (M) the length of M is (a) M (b) M (c) M (d) M 8. In fig. length of is (a) 0 cm (b) 9 cm (c) 5 5 cm (d) 5 cm 6 X Maths
4 8 cm 4 cm 6 cm 3 cm 9. In, and are points on side and respectively such that and : = 3 :. If = 3.3 cm then = (a). cm (b) 4.4 cm (c) 4 cm (d) 5.5 cm 0. and are two equilateral triangles such that is the midpoint of. Ratio of the areas of triangles and is (a) : (b) : (c) 4 : (d) : 4. In,. In the figure the value of x is x x 3 x x 5 (a) (b) (c) 3 (d) 3 X Maths 7
5 . In, = 90, is the perpendicular bisector of then ar ar (a) (b) (c) 4 (d) 4 3. The altitude of an equilateral triangle, having the length of its side cmis (a) cm (b) 6 cm (c) 6 cm (d) 6 3 cm 4. The straight line distance between and is (a) 3 5 (b) 5 3 (c) 5 (d) 5 5. If in an isosceles right-angled triangle the length of the hypotenuse is 0 cm then the perimeter of the triangle is (a) 5 cm (b) 5 cm (c) 0 cm (d) 0 cm 8 X Maths
6 SHORT NSWR TYP USTIONS 6. In figure ~ P. If = 8 cm, P = 4cm = 6.5 cm, P =.8 cm, find and. P 7. In the adjoining figure find if 4 cm 8. In the figure name the similar triangle. 3 cm x cm 8 cm P 0 cm 47 5 cm 47 cm X Maths 9
7 9. n isosecles triangle is similar to triangle PR. = = 4 cm, R = 0 cm and = 6 cm. What is the length of PR? Which type of triangle is PR? 0. In the figure ~ PR. What is the value of x? R In PR, R and R. Find 4 P P ar ar R x PR P. In triangles and PR if = and P R PR is the value of? R. then what 3. The measurement of three sides of a triangle are a, 0 a, 3 a. What is the measurement of the angle opposite to the longest side? 4. In the adjoining figure. What is the value of. 30 X Maths
8 0 cm cm 3 cm LONG NSWR TYP USTIONS 5. In the figure find SR if PR = PSR. PR = 6 cm and R = 9 cm P S 9 cm 6. In PR, RS P, RS = P, PS = 5 cm, SR = 8 cm. Find P. 7. Two similar triangles and P are made on opposite sides of the same base. Prove that = P. 8. In a quadrilateral, = 90, = + +. Prove that = cm R X Maths 3
9 9. In figure, = 3 cm, = 9 cm and ar () = 30 cm. Find ar (trap. ). 3 cm 9 cm 30. mit is standing at a point on the ground 8m away from a house. mobile network tower is fixed on the roof of the house. If the top and bottom of the tower are 7m and 0m away from the point. Find the heights of the tower and house. 3. In a right angled triangle, right angle at, 3. Find. 3. In a right angled triangle PRO, PR is the hypotenuse and the other two sides are of length 6cm and 8cm. is a point outside the triangle such that P = 4cm R = 6cm. What is the measure of PR? 33. In the figure is isosceles with =, P is the mid point of. If PM and PN. Prove that MP = NP. M N P 3 X Maths
10 34. PRS is a trapezium. S is a diagonal. and F are two points on parallel sides P and RS respectively intersecting S at G. Prove that SG = G SF. 35. Two poles of height a metres and b metres are apart. Prove that the height of the point of intersection of the lines joining the top of each pole ab to the foot of the opposite pole is given by a b mts. bm O h x L y 36. Show that the areas of two similar triangles are in the ratio of the squares (of the corresponding angle bisector segments). 37. In a rhombus, prove that four times the square of any sides is equal to the sum of squares of its diagonals. 38. is a trapezium with. If is similar to. Prove that =. 39. In a triangle, if the square of one side is equal to the sum of the squares on the other two sides, then prove that the angle opposite to the first side is a right triangle. 40. Prove that in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. 4. is a rectangle in which length is double of its breadth. Two equilateral triangles are drawn one each on length and breadth of rectangle. Find the ratio of their areas. 4. mar and shok are two friends standing at a corner of a rectangular garden. They wanted to drink water. mar goes due north at a speed of 50m/min and shok due west at a speed of 60m/min. They travel for 5 am X Maths 33
11 minutes. mar reaches the tap and drink water. How far (minimum distance) is shok from the tap now. 43. If two triangles are equiangular, prove that the ratio of the corresponding sides is same as the ratio of the corresponding altitudes. 44. In figure,if and, prove that is a right triangle. 45. In figure and : = 5 : 4. Find ar F ar F F 34 X Maths
12 NSWRS. c. b 3. a 4. d 5. c 6. d 7. a 8. c 9. b 0. c. d. d 3. d 4. a 5. c 6. = 5.6 cm, = 3.5 cm 7..5 cm 8. P ~ 9. 0 cm cm. 6 : cm 5. 4 cm cm cm 30. 9m, 6m : m cm 45. ar ar F F 5 8 X Maths 35
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