EWHO-2116

1.The internal angles of a polygon are in arithmetic

10. In the following figure
Progression. The smallest angle measures 65 0 and the
ABCDF is a regular
common difference is 18 ¾0 . The number of sides of the
pentagon and EDC is an
polygon is
equilateral triangle. Find the
A) 8 B) 9 C) 10 D) 7
measure of angle DFE.

2. In a polygon of n-sides the number of concave angle is 7,

find the minimum possible values of ‘n’.

11. In the following figure

ABCDE is a pentagon. Side AE,
3. What is the ED, DC and CB are of length
value of x? 10cm, 5cm, 6cm and 4cm
respectively. Find the length of
side AB.

4. In the given figure, angle EBG is twice of angle BEC and

AB is parallel to CD. Find ∟EBG . 12.ABCDE is a pentagon, the measure of each of the angle A,
B, C, D and E is 120 degree. The side AB is parallel to ED,
Find the difference between the length of side AB and CD if it
is known that AE and ED are of length 12 cm and 20 cm
respectively.

5. In the figure, AB,

CD and EF are
parallel lines. PQ and 13.
RS are transversals
cutting these parallel
lines are shown. If XY
= 5cm, YZ=9cm and
UW=21cm, find VW.

6. Find the sum

of all angles from
A to E

7. Find the sum of all angles

14. The measure of each exterior angle of a regular polygon
from A to J. is 45o then find the number diagonals that can be drawn inside
the polygon.

15. In a Decagon how many triangles , using the vertices of a

Decagon can be formed such that only one side of the triangle
is same as one side of the Decagon.

16. There are 15 iron rods of distinct integral length, find the
minimum possible length of the longest rod so that no triangles
can be formed with the help of any three rods out of the 15
8. The magnitude of greatest angle of a convex polygon rods.
having ‘n’ sides is 144o. Find the maximum possible value of
‘n’. If it is known that all the interior angles are of distinct 17. It is known that the measure of all the angles of a triangle
integral magnitude. (in degrees) are integers. If one of the angle of triangle is 40 o,
how many combinations of the angles of the triangle are
9. In a polygon, sides are either parallel to x-axis or y- axis . If
possible if the triangle is (i) acute? (ii) scalene?
the angle between two sides is 90 degrees , we say it’s a
convex angle. There are 25 such convex angles. If the angle
between two sides is 270 degrees , we say it’s a concave 18. A scalene triangle has integral sides and a perimeter of
angle. How many concave angles are there in the polygon. 24. How many such triangles are possible?
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19. In the following triangle ABC, x + y = 35 and u + v = 37. Find the value of u.

20. In the following figure, ABC is an isosceles triangle having side AB and BC equals to 16 unit. D is a point on
BC such that AD is 12 cm. Find the number of possible integral pairs of a and b.

22. Six regular hexagon surround a regular hexagon

of side length 1 as shown. What is the area of triangle
ABC.

23. Find the ratio between the longest diagonal and the shortest diagonal of a regular octagon.

24.Consider an obtuse-angled triangle whose sides are 8cm, 15cm and x cm where x is an integer. Find the number of
possible values of x.

25. The difference between the sum of length of any two sides with respect to third side in cyclic order is 8cm, 10cm and
12 cm. Find the area of triangle.

26. The altitude from each of the vertex of a triangle is drawn to the opposite side. The length of altitudes are in the ratio
3 : 5 : 8. Find the minimum possible perimeter of triangle provided that sides are of integral length.

27. If the angles of a triangle are 60o and 45o, and the included side is (2+√3)cm then find the area of the triangle.

28. Find the percentage

shaded portion
29.What percentage of triangle is shaded?

30. In the following triangle, AD is a median and

AE:EB =2:5. Find the ratio of line segment EP:PC.

31.Area of triangle ABC is 60 square units. If the shaded

and unshaded regions have equal area, find the area of
quadrilateral PQRS in square units.

32. Find the area of quadrilateral ADPE.

33. The area of two shaded portion inside the rectangle is 8 unit and 3 unit respectively. Find the area of rectangle
ABCD.

34. Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square
whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is………..
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35. G is the centroid of the triangle PQR. PQ =48cm, QR=55cm and PR=73cm. S is the mid-point of PR. Find the measure
of GS.

36. The length of three medians of a triangle ABC is 10cm, 11cm and 13cm. Find the sum of the squares of three sides
of the triangle.

37. The medians of a triangle ABC are of length 18cm, 24cm and 30 cm respectively. Find the area of triangle ABC.

38. The bisectors of angle X and angle Y in a triangle XYZ meet at I. If ∠YIZ = 125o, find the measure of ∠YXZ. The
bisectors of exterior angle of Y and exterior angle Z meet at P. Find the measure of angle YPZ.

39. In the following triangle ABC the length of side AB,

BC and AC is 6cm, 7cm and 8cm respectively. AD and
BE are angle bisectors of ∠BAC and ∠CBA meets at
point I. Find (i) the length of AI (ii) the percentage
shaded portion.

40. ABC is a right angled triangle LB = 90 0. AB = 24 BC = 7. Point D lies on the extended line BC and AD is an external
angular bisector. Find CD.

41. ABC is a right angled triangle LB = 900. AB = 8 BC = 6. Point D lies on the extended line BC and AD is an external
angular bisector. Find AD.

42. Consider triangle ABC with AB=8cm, BC=7cm and CA=6cm. Point E lies on the extended line BC. AD and AE are
internal and external bisectors of Angle A, what is AE 2 + AD2 ?

43. ABC is triangle having side AB and AC are of side length 17.5cm and 9cm respectively. D is a point on side BC such
that AD is perpendicular on side BC and is of length 3cm. Find the circumradius of triangle ABC.

44. There are 12 equidistant points around the circle. How many right-angled triangle can be formed with the help of these
12 points.

45. One of the side of an obtuse angled isosceles triangle of perimeter 98cm is 48cm long. What is the distance between
the circumcentre and incentre of triangle?

46. The inradius and circumradius of a right angled triangle is 8cm and 20cm. Find the perimeter of the triangle.

47. P is the orthocentre of an acute-angled triangle ABC. The measure of angle BPC is 130o. Find the measure of angle
BAC.

48. ABC is a right angle triangle having angle A = 90 degree and length of side BC is 70 cm. Point X and Y lies on
the side BC such that AX is angle bisector and AY is median. The distance between point X and Y is 5 cm. Find
the area of triangle ABC in(sq cm).
A. 1226 B.1356 C. 1176 D.1484
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49. Given that ∠ABH = 90o and AB is parallel to CD, EF and GH. Find the
length of CD and EF, if the length of AB and GH is 10cm and 15cm
respectively.

50. Consider the triangle ABC shown in the following figure where BC = 12 cm, DB = 9 cm, CD = 6 cm and ∠BAC
= ∠BCD. Find the perimeter of triangle ADC.

51. In the figure given below, P is a point inside the triangle ABC,
Line segments DE, FG and HI are drawn through P, parallel to the
sides AB, BC and CA respectively. The area of three triangles DPG,
FPI and EPH are 1,9 and 25 respectively. What is the area of the
triangle ABC?(all the areas are in sq. cm)

52. In the adjoining triangle ABC. These 20 equidistant parallel lines B1 C1, B2 C2, ……., B19 C19, BC. Find the
ratio of area of region x & y?

53. In the adjoining triangle ABC, DE is parallel to FG and FG is parallel to BC such that Area of triangle ADE,
Area of quadrilateral DEGF and Area of Quadrilateral FGCB are equal. Find the ratio between h 1, h2 and h3.

54. In the adjoining triangle ABC, DE is parallel to BC and area of triangle PDE is 3/7 of area of triangle ADE.
Find the ratio of DE to BC.
55. In the adjoining figure, CD is the diameter of circle and chord AB is parallel to CD. The measure of angle AEB
is 40o . What is the measure of angle ABC?

56. .In the following figure, AC is diameter of the circle and parallel to ED and the measure of ∠CBE = 65o . Find
the measure of ∠DEC.

57. Find the measure of angle(i) CPB (ii) ABF

58. In the adjoining figure angle BAD is 65o. Find the measure of angle CAO.

59. . In the following figure, PAB is an equilateral triangle. The measure of angle DCB is 20o and the length of DB

is 7cm, find DC(in cm).

60. Two non-equal circles touch each other at M. AC is direct common tangent of the two circles. AB and CD are
chords. If angle ABM is 30 degree, what is the measure of angle CMD?

61. In the following figure, The measure of angle BAC is 90o and the length of EF is 15cm. Find the length of AC.

62. In the figure given below, ABC is a triangle, D and E are the midpoints of side BC and AC respectively. Find
the ratio of length PD and EC.

63. Find the area of triangle PQR, it is known that the measure of angle PSR is 90 degree.

64. A circle is divided into 12 equal parts by the points A1,A2,A3, A4 , …….A10,A12 as shown in the fig. Find the
ratio of the measure of angle P to angle Q
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65. Two circles, one with centre A and one with centre B, intersect at points P and Q such that ∠P AQ = 60◦ and ∠P BQ
= 90◦ . What is the ratio of area of the circle with centre A to the area of the circle with centre B?

66. MN is a transverse tangent. The radius of circle with center P and Q is 15cm and 9cm respectively. The distance
between the center of two circle is 40cm. (i) Find the length of BC. (ii) Find the length of MN.(iii) Find the length of MC.

67. The radius of circle having center P and Q is 16cm and 9cm respectively. Find the radius of small inscribed circle
between two circles and their direct common tangents.

68. In the adjoining figure, CAB = 900 and the radius of circle is 5 cm. Find the perimeter of triangle
ABC.

69. Three equal circles of unit radius have a common tangent AB. Find the radius of semicircle.
70. Two circles inscribed in a triangle, touches each other at point E. AD is the common tangent of both circles.
AB = 12, BC = 18 & AC = 10. Find the length of CD.

71. Two circles are circumscribed by third circle. AB is common tangent of two smaller circles and chord of large
circle, If AB = 16 cm. Find the shaded area.

72. Find the radius of inscribed circle.

73. Two circle of radius 1 unit passes through the centre of each other. Find the area of their common region.

74. A circle has its centre O. A and B are two points on the circle. The area of the minor sector AOB is 48π sq. cm. If the
length of arc AOB is 8π cm, find the radius of the circle.

75. Three circles passes through the centre of each other. Find the perimeter of the figure and the area of common region
of all three circles.

76. In the given figure, AB = 6 cm and O is the middle point of AB. Semicircles are drawn on AB, AO and OB. If C is the
center of the small circle which touches all the semicircles, then the radius of this small circle is

77. In the following diagram, three circles of unit radius passes through the centre of fourth circle, Find the
area of the shaded portion
78. In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P
be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is …….

79. In a parallelogram, lengths of two diagonals is 12cm and 2 46 cm. If the difference between the lengts of two
sides is 2cm, find its area.

80.In the following figure, ABCD is a parallelogram. Find the value of x.

81. In a parallelogram ABCD, AB = 20cm,AD = 12cm and DAB = 60o .Find the area of quadrilateral bounded
by the angle bisector of parallelogram ?

82. In a parallelogram PQRS, A and B are mid points of sides QR and RS respectively. If area of triangle PQR is
12 square cm, find the area of triangle PAB ( in cm2)
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83. ABCD is a rectangle with perimeter 40cm. It is divided in to 8 congruent rectangles. Find area of ABCD

84. The length of rectangle is 17 unit more than the breadth and the diagonal is 73 unit. Find the area of rectangle.

85. In the diagram, ABCD is a rectangle. Point P is located inside the rectangle so that the distance from P to A is 5 cm,
the distance from P to B is 11 cm, and the distance from P to D is 10 cm. How far is P from C?

86. The radius of the circle inscribed the rectangle ABCD is 1 unit. The centre of the quarter circle is at B. Find the area
of rectangle.

87. In the following figure, the length and breadth of the rectangle is 12cm and 5 cm respectively.
There are two inscribed circles between diagonal line AC and the two sides of rectangle. Find the
length of the line segment EF.

88. In the following figure, rectangle ABCD is placed in such a manner that side BC is tangent to circle
and vertex A and B lies on the circle. The length and breadth of the rectangle is 12cm and 8 cm
respectively. Find the radius of circle.
89. Three squares are placed beside each other as shown. The smallest square has side length 4 units, the middle-sized
square has side length 7 units, but the side length of the largest square is unknown. However, the top left corner of each
of the three squares lies on a straight line. Determine the side length of the largest square.

90. ABCD is a square of side length 12 unit. What is the length of each side of small square PQRS inscribed between
circle and bigger square ABCD?

91. ABCD is a square. E is the midpoint of BC. Find the ratio of inradius of the circle inscribed in the triangle DCE, to
the side of the square.

92. In the figure, ABC and GFH are equilateral triangles, and DEFG is a square. Find the measure of angle GAD and
HDA.

93. The area of a rhombus having side 8cm is 36cm2 . Find the length of bigger diagonal.
94. In the figure, ABCD is a rhombus and ∠ACB = 30o . If the diagonal AC is x times the diagonal DB, what is the value
of x?

95. In the figure, O is the centre of circle and OABC and OEDC are congruent rhombuses. If the total area of two
rhombuses is 64√3 sq.cm, the radius of circle is ………….

96. If the length of two diagonals of a rhombus is 12cm and 16cm. Find the measure of its inradius.

97. In the following figure AB and DC are parallel. If AB : DC =1 : 4 and area of triangle ABE is 10 sq.unit, find the area
of trapezium.
98. In trapezium BCED, DE is parallel to BC. DE=5cm, DB=25cm and BC=30cm. If the height of the trapezium is 24cm,
what is the perimeter of trapezium BCED?

99 ABCD is a trapezium such that AB || CD. EF is any line segment parallel to AB. AB is the longer parallel side of the trapezium.
Length of AB is 7 units more than DC and DE : EA = 2 : 5. The length of line segment EF is 9 units. Find the length of AB.

100. Find the area of trapezium whose parallel sides are of length 60 unit and 100 units and the non-parallel sides are of
length 20 unit and 30 unit respectively.

101.If the sides AB, BC, CD and DA of trapezium ABCD measure 10cm, 20 cm, 18cm and 16cm respectively, find the
length of the longer diagonal, given that AB is parallel to CD.

102. PQRS is a cyclic quadrilateral. Find the sum of angles A, B,C and D.

103. ABCD is a cyclic quadrilateral having angle A is equal to 120 o and side AD is of length 5 unit. CD is the diameter of
circle which is parallel to AB. Find the area of cyclic quadrilateral.

104. In the following figure O is

the centre of circle having
diameter 65 unit. DA and BC
are of length 25 unit and 39unit
respectively. Find the area of
Cyclic quadrilateral ABCD.

105. Find the minimum area of the following rectangle

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105. In the following figure, ABCDEF is a regular hexagon A circle circumscribe the regular hexagon so that the diameter
CF is extended to point P, PF and PA are two tangents are drawn from point P. Find the ratio between area of trapezium
PEDC and area of circumscribed circle.

106. A regular polygon has an even number of sides, if the product of the length of its sides and the distance between the
opposite sides is ¼ of its area, find the number of sides it has.

107. If the area enclosed between circumcircle and incircle of a regular polygon of n sides is 36π sq. unit, then find the
value of n.

108. Four isosceles right-angled triangles are cut from each corner of the square of side 2cm so that the remaining part
is a regular octagon. Find the area and side length of regular octagon.

109. Three rings formed around a regular pentagon (R1), a equilateral triangle (R2), and a regular hexagon (R2) . Area
of R1, R2 and R3 are in the ratio of 36 : 49 : 25.If the perimeters of pentagon, equilateral triangle and hexagon are P1 ,P2
and P3,find P1 : P2 : P3.

110. In the given figure there are three regular hexagon with side 6√3 cm. What is the shortest distance between point
A and point B?

Mensuration-3D
111. In a convex polyhedron the number of faces is 12 and the number of vertex is 10, Find the number of edges.

112. A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1:1:8: 27:27. The percentage
by which the sum of the surface areas of this 5 cubes exceeds the surface area of the original cube is nearest to
A)10 b)50 c)60 d)20

113. The sum of three mutually perpendicular sides of a cuboid is 60cm and the diagonal is 40cm, Find the total surface
area of the cuboid.

114. The sum of the reciprocal of length and breadth of a cuboid shaped room is 5 unit. The magnitude of area of all 4
walls is how many times that of the volume of room.
115. If the rectangular faces of a brick have their diagonals in the ratio 3 : 2√3 : √15, then the ratio of the length of the
shortest edge of the brick to that of its longest edge is
116. A beetle walks on the surface of the 2×3×12 rectangular prism shown. The beetle wishes to travel from P to Q. What
is the length of the shortest path from P to Q that the beetle could take?

117. A rectangular swimming pool is 48 m long and 20m wide. The shallow edge of the pool is 1 m deep. For every 2.6m
that one walks up the inclined base of the swimming pool, one gains an elevation of 1m. What is the volume of water (in
cubic meters), in the swimming pool? Assume that the pool is filled up to the brim.

118. The rectangular base of an aquarium is 40 cm by 60 cm, and its height is 30 cm. The aquarium is tilted along AB
until the water completely covers the end ABCD. At this point, it also covers 4/5 of the base. Determine the depth of the
water, in centimetres, when the aquarium is level.

119. A well in the cylindrical shape is dug up in the ground. The whole soil dug out is spread around the boundary of the
pool uniformly to raise in the form of a thick wall. If the inner, and outer radii of the raised wall are 3 & 4 m, find the %
increase the volume of the well due to the raise cylindrical wall.

120. Find the radius of the largest possible sphere which can be placed inside the right circular cone of base radius 5cm
and vertical height 12cm.

121. A cylinder is placed inside the right circular cone of base radius 4cm and the vertical height 10cm in such a manner
that the flat surface area of the cylinder is resting on the base of cone, Find the maximum total surface area of the cylinder.