## Chapter 14: GEOMETRY

### Introduction

• Sum of the angles of a triangle is 180°.
• Sum of any two sides of a triangle is greater than the third side.
• Area of a rectangle = length * breadth.
• Area of a square = side^2 = 1/2(diagonal)^2
• Area of a circle = π * r*r ; r = radius and π / PI = 22/7 or 3.14
• Area of a kite: 1/2 * (sum of the length of two diagonals)
• Area of a semi circle = π * r * r / 2 ; Circumference = π * r
• To find the area of a parallelogram, just multiply the base length (b) times the height (h): Area = b * h

### Volume and Surface Area

Cuboid: If length = L, breadth = B, height = H then Volume = L * B * H.

volume of a cube = a3 , Surface area = 6 *a * a sq m, diagonal = sq-root(3) * a

cylinder = b h = π *  r2

pyramid = (1/3) b

volume of a cone = (1/3) b h = 1/3 * π *r2

volume of a sphere = (4/3) * π *  r3

1 m^3 = 1000 liters ; 1 cm^3 = 1 liter

### CAT Problems for practice

Q. In △ABC, the internal bisector of ∠ A meets BC at D. If AB = 4, AC = 3 and , then the length of AD is

1. 2√3

2. 12 √3 /7

3. 15 √3 /8

4. 6 √3 /7

Ans . B

1. Area of △ ABC = Area of △ ABD + Area of △ ADC

2. 1/2 * 4 * 3sin 60° = 1/2 * 4 * ysin 30° + 1/2 * 3y * sin 30°

3. 12 √3 = 4y + 3y

4. y = 12√3 / 7

Q. Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be

1. 85.5

2. 92.5

3. 90.5

4. 87.5

Ans . D

1. The surface area of a sphere is proportional to the square of the radius.

2. The SB / SA = 4/1. Surface area of B is 300% higher than A)

3. RB / RA = 2/1. The volume of a sphere is proportional to the cube of the radius.

4. Thus VB / VA = 8/1. Thus VA is 7/8th less than VB which is 87.5%

Q. The length of the common chord of two circles of radii 15 cm and 20 cm, whose centres are 25 cm apart, is

1. 24 cm

2. 25 cm

3. 15 cm

4. 20 cm

Ans . A

1. Let the length of the chord be x cm.

2. $\frac{1}{2}(15 * 20) = \frac{1}{2} * (25) *\frac{x}{2}$

3. x = 24

Q. In a 4000 meter race around a circular stadium having a circumference of 1000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5 th minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, what is the time taken by the fastest runner to finish the race?

1. 20 min

2. 15 min

3. 10 min

4. 5 min

Ans . C

1. The ratio of the speeds of the fastest and the slowest runners is 2 : 1. Hence they should meet at only one point on the circumference i.e. the starting point (As the difference in the ratio in reduced form is 1). For the two of them to meet for the first time, the faster should have completed one complete round over the slower one. Since the two of them meet for the first time after 5 min, the faster one should have completed 2 rounds (i.e. 2000 m) and the slower one should have completed 1 round. (i.e. 1000 m) in this time. Thus, the faster one would complete the race (i.e. 4000 m) in 10 min.

Q. Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be

1. 20

2. 0

3. 21

4. 22

Ans . C

1. Let the total number of angles in the polygon be x. Therefore, the number of concave corners will be x – 25.

2. 25 × 90 + (x – 25) × 270 = (x – 2) × 180 ⇒ x = 46

3. ⇒ x – 25 = 21.

Q. Four horses are tethered at four corners of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.

1. 22

2. 42

3. 84

4. 168

Ans . A

1. Total area = 14 × 14 = 196 m2

2. Grazed area = $\frac{π * r2}{4} * 4 3. Grazed area = 154 4. Ungrazed area is less than (196 – 154) = 42 m2, for which there is only one option i.e. 22 m2 Q. In the figure given below, ABCD is a rectangle. The area of the isosceles right triangle ABE = 7 cm2; EC = 3(BE). The area of ABCD in cm2 is 1. 21 2. 28 3. 42 4. 56 Ans . D 1. Area of ∆ABE = 7 cm2 2. Area of rectangle ABEF = 14 cm2 3. Area of ABCD = 14 × 4 = 56 cm2 Q. The area of the triangle whose vertices are (a, a), (a + 1, a + 1) and (a + 2, a) is 1. a3 2. 1 3. 2a 4. √ 2 Ans . B 1. Let a = 0 2. Hence, area =$\frac{1}{2} * 2 * 1\$ = 1

3. Answer should be independent of 'a' and area of the triangle doesn't have square root

Q. Instead of walking along two adjacent sides of a rectangular field, a boy took a short cut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is

1. 1/2

2. 2/3

3. 1/4

4. 3/4

Ans . D

1. With option 3/4, diagonal is 5.

2. Distance saved = (4 + 3) – 5 = 2 = Half the larger side.

Q. Neeraj has agreed to mow a lawn, which is a 20 m × 40 m rectangle. He mows it with 1 m wide strip. If Neeraj starts at one corner and mows around the lawn toward the centre, about how many times would he go round before he has mowed half the lawn?

1. 2.5

2. 3.5

3. 3.8

4. 4

Ans . C

1. Area = 40 × 20 = 800 m2

2. If 3 rounds are done, area = 34 × 14 = 476 m2

3. Area > 3 rounds ; If 4 rounds ⇒ Area left = 32 × 12 = 347 m2

4. Hence, area should be slightly less than 4 rounds.

Q. Ten straight lines, no two of which are parallel and no three of which pass through any common point, are drawn on a plane. The total number of regions (including finite and infinite regions) into which the plane would be divided by the lines is

1. 56

2. 255

3. 1024

4. not unique

Ans . A

1. Number of regions = n(n+1)/2 + 1

2. where n = Number of lines, i.e. for 0 line we have region = 1. For 1 line we have region = 2. It can be shown as:

3. Therefore, for n = 10, it is ( 10 * 11 )/2 + 1 = 56

In the diagram ∠ ABC = 90° = ∠ = ∠ DCH = ∠ DOE = ∠ EHK = ∠ FKL = ∠ GLM = ∠ LMN; AB = BC = 2CH = 2CD = EH = FK = 2HK = 4KL = 2LM = MN

Q. The magnitude of ∠ FGO =

1. 30

2. 45

3. 60

4. none

Ans . D

1. if KL = 1, then IG = 1 and FI = 2

2. hence tan θ = 2/1 = 2

3. thus θ cant be 30 / 60 /90.

Q. What is the ratio of the areas of the two quadrilaterals ABCD to DEFG?

1. 1 : 2

2. 2 : 1

3. 12 : 7

4. None of these

Ans . C

1. Area of quadrilateral ABCD = 1/2 * (2x+4x) * 4x = 12x

2. Area of quadrilateral DEFG = 1/2 * (5x+2x) * 2x = 7x

3. Hence, ratio = 12 : 7

Q.Let the consecutive vertices of a square S be A, B, C & D. Let E, F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to

1. 1/2
2. 1/3
3. 1/4
4. 1/8

Ans.a

Q.A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. 2 : 3
2. 3 : 4
3. 1 : 4
4. 1 : 2

Ans.d

Q.A square piece of cardboard of sides ten inches is taken and four equal squares pieces are removed at the corners, such that the side of this square piece is also an integer value. The sides are then turned up to form an open box. Then the maximum volume such a box can have is

1. 72 cubic inches.
2. 24.074 cubic inches.
3. 2000/27 cubic inches
4. 64 cubic inches.

Ans.a

Q.Three identical cones with base radius r are placed on their bases so that each is touching the other two. The radius of the circle drawn through their vertices is

1. smaller than r
2. equal to r
3. larger than r
4. depends on the height of the cones.

Ans.c

Q.The line AB is 6 metres in length and is tangent to the inner one of the two concentric circles at point C. It is known that the radii of the two circles are integers. The radius of the outer circle is

1. 5 metres
2. 4 metres
3. 6 metres
4. 3 metres

Ans.a

Q.Four cities are connected by a road network as shown in the figure. In how many ways can you start from any city and come back to it without travelling on the same road more than once?

1. 8
2. 12
3. 16
4. 20

Ans.b

Q.A slab of ice 8 inches in length, 11 inches in breadth, and 2 inches thick was melted and resolidified into the form of a rod of 8 inches diameter. The length of such a rod, in inches, is nearest to

1. 3
2. 3.5
3. 4
4. 4.5

Ans.b

Q.Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points?

1. 4
2. 6
3. 8
4. 10

Ans.c

Q.The diameter of a hollow cone is equal to the diameter of a spherical ball. If the ball is placed at the base of the cone, what portion of the ball will be outside the cone?

1. 50%
2. less than 50%
3. more than 50%
4. 100%

Ans.c

Q.A right circular cone, a right circular cylinder and a hemisphere, all have the same radius, and the heights of the cone and cylinder are equal to their diameters. Then their volumes are proportional, respectively to

1. 1 : 3 : 1
2. 2 : 1 : 3
3. 3 : 2 : 1
4. 1 : 2 : 3

Ans.a

Q.A right circular cone of height h is cut by a plane parallel to the base and at a distance h/3 from the base, then the volumes of the resulting cone and the frustum are in the ratio

1. 1 : 3
2. 8 : 19
3. 1 : 4
4. 1 : 7

Ans.b

### Read and solve the questions

Q.ABCD is a square of area 4, which is divided into four non-over lapping triangles as shown in figure. Then the sum of the perimeters of the triangles is

1. 8(2 + √2)
2. 8(1+ √2)
3. 4(1 + √2)
4. 4(2 + √2)

Ans.b

Q.AB ⊥ BC, BD ⊥ AC and CE bisects ∠ C, ∠ A = 30°. Then what is ∠ CED?

1. 30
2. 60
3. 45
4. 65

Ans.b

Q.PQRS is a square. SR is a tangent (at point S) to the circle with centre O and TR = OS. Then the ratio of area of the circle to the area of the square is

1. 1/3
2. 11/7
3. 7/11
4. 3

Ans.a

Q.In the adjoining figure, AC+ AB = 5AD and AC – AD = 8. Then the area of the rectangle ABCD is

1. 36
2. 50
3. 60
4. cant say

Ans.c

Q.In the given figure, AB is diameter of the circle and points C and D are on the circumference such that ∠ CAD = 30° and ∠ CBA = 70°. What is the measure of ∠ ACD?

1. 40
2. 50
3. 30
4. 90

Ans.a

Q.The length of a ladder is exactly equal to the height of the wall it is learning against. If lower end of the ladder is kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then the height of the wall is

1. 12
2. 15
3. 18
4. 11

Ans.b

Q.The sides of a triangle are 5, 12 and 13 units. A rectangle is constructed, which is equal in area to the triangle, and has a width of 10 units. Then the perimeter of the rectangle is

1. 30
2. 36
3. 13
4. none

Ans.d

Q.A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?

1. 16

2. 8

3. 12

4. 24

Ans.b

Q.If ABCD is a square and BCE is an equilateral triangle, what is the measure of ∠DEC?

1. 15

2. 30

3. 20

4. 45

Ans.a

Q.From a circular sheet of paper with a radius 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the uncut to the cut portion?

1. 1 : 3

2. 4 : 1

3. 3 : 1

4. 4 : 3

Ans.c

Q.A wooden box (open at the top) of thickness 0.5 cm, length 21 cm, width 11 cm and height 6 cm is painted on the inside. The expenses of painting are Rs. 70. What is the rate of painting per square centimetres?

1. Re. 0.7

2. Re. 0.5

3. Re. 0.1

4. Re. 0.2

Ans.c

Q.The figure shows a circle of diameter AB and radius 6.5 cm. If chord CA is 5 cm long, find the area of Triangle ABC.

1. 60

2. 30

3. 40

4. 52

Ans.b

Q.In the adjoining figure, points A, B, C and D lie on the circle. AD = 24 and BC = 12. What is the ratio of the area of △CBE to that of △ADE?

1. 1/4

2. 1/2

3. 1/3

4. insufficient data

Ans.a

Q.In the given figure, EADF is a rectangle and ABC is a triangle whose vertices lie on the sides of EADF and AE = 22, BE = 6, CF = 16 and BF = 2. Find the length of the line joining the mid-points of the sides AB and BC.

1. 4√2

2. 5

3. 3.5

4. none

Ans.b

Q.The value of each of a set of coins varies as the square of its diameter, if its thickness remains constant, and it varies as the thickness, if the diameter remains constant. If the diameter of two coins are in the ratio 4 : 3, what should be the ratio of their thickness if the value of the first is four times that of the second?

1. 16/9

2. 9/4

3. 9/16

4. 4/9

Ans.b

Q.In △ ABC, points P, Q and R are the mid-points of sides AB, BC and CA respectively. If area of △ ABC is 20 sq. units, find the area of △ PQR. in sq. units

1. 10

2. 5√3

3. 5

4. none

Ans.c

Q.In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of the longer side. What is the ratio of the shorter to the longer side?

1. √3 : 2

2. 1 : √3

3. 2:5

4. 3:4

Ans.d

Q. Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of △ PQR?

1. 120

2. 66

3. 93

4. 87

Ans . C

A cow is tethered at point A by a rope. Neither the rope nor the cow is allowed to enter △ ABC. l(AB) = l(AC) = 10m. ∠ BAC = 30.

Q. What is the area in sq meters that can be grazed by the cow if the length of the rope is 8 m?

1. 134 1/3

2. 121

3. 132

4. 176/3

Ans . D

Q. What is the area that can be grazed by the cow if the length of the rope is 12 m?

1. 133 1/3

2. 121

3. 132

4. 176/3

Ans . A

Q. Four identical coins are placed in a square. For each coin the ratio of area to circumference is same as the ratio of circumference to area. Then find the area of the square that is not covered by the coins.

1. 16(4-1)

2. 16(8-1)

3. 16(4-2)

4. 16(2)

Ans . A

Q. Ten points are marked on a straight-line and 11 points are marked on another straight-line. How many triangles can be constructed with vertices from among the above points?

1. 495

2. 550

3. 1045

4. 2475

Ans . C

Q. There is a square field of side 500 m long each. It has a compound wall along its perimeter. At one of its corners, a triangular area of the field is to be cordoned off by erecting a straight-line fence. The compound wall and the fence will form its borders. If the length of the fence is 100 m, what is the maximum area that can be cordoned off?

1. 2,500 sq m

2. 10,000 sq m

3. 5,000 sq m

4. 20,000 sq m

Ans . A

### Answer the questions based on the following information

A rectangle PRSU, is divided into two smaller rectangles PQTU, and QRST by the line TQ. PQ = 10 cm. QR = 5 cm and RS = 10 cm. Points A, B, F are within rectangle PQTU, and points C, D, E are within the rectangle QRST. The closest pair of points among the pairs (A, C), (A, D), (A, E), (F, C), (F, D), (F, E), (B, C), (B, D), (B, E) are 10 √3 cm apart.

Q. Which of the following statements is necessarily true?

1. The closest pair of points among the six given points cannot be (F, C)

2. Distance between A and B is greater than that between F and C.

3. The closest pair of points among the six given points is (C, D), (D, E), or (C, E).

4. None of the above

Ans . A

Q. AB > AF > BF ; CD > DE > CE ; and BF = 6√5cm. Which is the closest pair of points among all the six given points?

1. B, F

2. C, D

3. A, B

4. None of these

Ans . D

### Answer the questions based on the following information.

Q. In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straight- line segments, shown as solid lines, in the domain x ∈ (-2 , 2)

1. if F1(x) = –F(x)

2. if F1(x) = F(–x)

3. if F1(x) = –F(–x)

4. if none of the above is true

Ans . D

Q. In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straight- line segments, shown as solid lines, in the domain x ∈ (-2 , 2)

1. if F1(x) = –F(x)

2. if F1(x) = F(–x)

3. if F1(x) = –F(–x)

4. if none of the above is true

Ans . B

Q. In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straight- line segments, shown as solid lines, in the domain x ∈ (-2 , 2)

1. if F1(x) = –F(x)

2. if F1(x) = F(–x)

3. if F1(x) = –F(–x)

4. if none of the above is true

Ans . B

Q. In each of the following questions, a pair of graphs F(x) and F1(x) is given. These are composed of straight- line segments, shown as solid lines, in the domain x ∈ (-2 , 2)

1. if F1(x) = –F(x)

2. if F1(x) = F(–x)

3. if F1(x) = –F(–x)

4. if none of the above is true

Ans . C

Q. A farmer has decided to build a wire fence along one straight side of his property. For this, he planned to place several fence-posts at 6 m intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was 5 less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them 8 m apart. What is the length of the side of his property and how many posts did he buy?

1. 100 m, 15

2. 100 m, 16

3. 120 m, 15

4. 120 m, 16

Ans . D

Q. ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and stays there. Let an be the number of distinct paths of exactly n jumps ending in E. Then what is the value of a2n-1

1. 0

2. 4

3. 2n-1

4. cant determine

Ans . A

Q. There are three cities: A, B and C. Each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly, there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?

1. 6

2. 3

3. 5

4. 10

Ans . A

Q. In the figure below, AB = BC = CD = DE = EF = FG = GA. Then ∠ DAE is approximately

1. 15

2. 20

3. 30

4. 25

Ans . D

Q. If a, b and c are the sides of a triangle, and a2 + b2 + c2 = bc + ca + ab, then the triangle is

1. equilateral

2. isosceles

3. right-angled

4. obtuse-angled

Ans . A

Q. The area bounded by the three curves |x + y| = 1, |x| = 1, and |y| = 1, is equal to

1. 4

2. 3

3. 2

4. 1

Ans . B

Q. Consider a circle with unit radius. There are seven adjacent sectors,S1, S2, S3, ...S7, in the circle such that their total area is 1/8 of the area of the circle. Further, the area of the jth sector is twice that of the (j – 1)th sector, for j = 2, ..., 7. What is the angle, in radians, subtended by the arc of S1 at the centre of the circle

1. 1/508

2. 1/2040

3. 1/1016

4. 1/1524

Ans . A

Q. ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

1. x + y = –1

2. x – y = –1

3. x + y = 1

4. None of these

Ans . A

Directions for questions Given below are three graphs made up of straight line segments shown as thick lines. In each case choose the answer as
(a) if f(x) = 3 f(–x)
(b) if f(x) = –f(–x)
(c) if f(x) = f(–x)
(d) if 3f(x) = 6 f(–x), for x ≥ 0

1. A

2. B

3. C

4. D

Ans . C

1. A

2. B

3. C

4. D

Ans . D

1. A

2. B

3. C

4. D

Ans . B

Q. What is the number of distinct triangles with integral valued sides and perimeter 14?

1. 6

2. 5

3. 4

4. 3

Ans . C

Q. The figure below shows the network connecting cities A, B, C, D, E and F. The arrows indicate permissible direction of travel. What is the number of distinct paths from A to F?

1. 9

2. 10

3. 11

4. none

Ans . B

Q. A rectangular pool 20 m wide and 60 m long is surrounded by a walkway of uniform width. If the total area of the walkway is 516 m2 , how wide, in metres, is the walkway?

1. 43

2. 4.3

3. 3

4. 3.5

Ans . C

Q. Based on the figure below, what is the value of x, if y = 10?

1. 10

2. 11

3. 12

4. none

Ans . B

Q. In DEF shown below, points A, B and C are taken on DE, DF and EF respectively such that EC = AC and CF = BC. If ∠ D = 40° , then ∠ ACB =

1. 140

2. 70

3. 100

4. none

Ans . C

Q. Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side

1. √ 260

2. √250

3. √240

4. √270

Ans . A

Q. Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles.

1. 768

2. 534

3. 696.5

4. 684

Ans . D

Q. A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

1. 10

2. 15

3. 20

4. 17

Ans . D

Q. In the above diagram, ABCD is a rectangle with AE = EF = FB. What is the ratio of the areas of △ CEF and that of the rectangle

1. 1/6

2. 1/8

3. 1/9

4. none

Ans . A

Q. A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?

1. 6

2. 9

3. 12

4. none

Ans . B

Q. A square, whose side is 2 m, has its corners cut away so as to form an octagon with all sides equal. Then the length of each side of the octagon, in metres, is

1. √2 / (√2+1)

2. 2 / (√2+1)

3. 2 / (√2-1)

4. √2 / (√2-1)

Ans . B

### Quiz

Score more than 80% marks and move ahead else stay back and read again!