CAT Quantitative Ability Questions | CAT Geometry questions

CAT/2023.3(Quantitative Ability)

Question. 1

CAT/2023.3(Quantitative Ability)

Question. 2

CAT/2023.3(Quantitative Ability)

Question. 3

In a regular polygon, any interior angle exceeds the exterior angle by 120 degrees. Then, the number of diagonals of this polygon is

CAT/2023.2(Quantitative Ability)

Question. 4

CAT/2023.2(Quantitative Ability)

Question. 5

In a rectangle ABCD, AB = 9 cm and BC = 6 cm. P and Q are two points on BC such that the areas of the figures ABP, APQ, and AQCD are in geometric progression. If the area of the figure AQCD is four times the area of triangle ABP, then BP : PQ : QC is

CAT/2023.2(Quantitative Ability)

Question. 6

CAT/2023.1(Quantitative Ability)

Question. 7

A quadrilateral ABCD is inscribed in a circle such that AB : CD = 2 : 1 and BC : AD = 5 : 4. If AC and BD intersect at the point E, then AE : CE equals

CAT/2023.1(Quantitative Ability)

Question. 8

CAT/2023.1(Quantitative Ability)

Question. 9

In a right-angled triangle ∆ABC, the altitude AB is 5 cm, and the base BC is 12 cm. P and Q are two points on BC such that the areas of ∆ABP, ∆ABQ and ∆ABC are in arithmetic progression. If the area of ∆ABC is 1.5 times the area of ∆ABP, the length of PQ, in cm, is

CAT/2022.3(Quantitative Ability)

Question. 10

Suppose the medians BD and CE of a triangle ABC intersect at a point O. If area of triangle ABC is 108 sq. cm., then, the area of the triangle EOD, in sq. cm., is

CAT/2022.3(Quantitative Ability)

Question. 11

The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is

CAT/2022.3(Quantitative Ability)

Question. 12

CAT/2022.2(Quantitative Ability)

Question. 13

CAT/2022.2(Quantitative Ability)

Question. 14

CAT/2022.2(Quantitative Ability)

Question. 15

CAT/2022.1(Quantitative Ability)

Question. 16

CAT/2022.1(Quantitative Ability)

Question. 17

Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4) and (−2, 8), respectively. Then, the coordinates of the vertex D are

CAT/2022.1(Quantitative Ability)

Question. 18

CAT/2021.1(Quantitative Ability)

Question. 19

Suppose the length of each side of a regular hexagon ABCDEF is 2 cm. If 'T' is the mid point of CD, then the length of AT, in cm, is:

CAT/2021.3(Quantitative Ability)

Question. 20

In a triangle ABC, ∠BCA = 50. D and E are points on AB and AC, respectively, such that AD=DE. If F is a point on BC such that BD=DF, then ∠FDE in degrees, is equal to

CAT/2021.3(Quantitative Ability)

Question. 21

Let ABCD be a parallelogram. The lengths of the side AD and the diagonal AC are 10 cm and 20 cm, respectively. If the angle ∠ADC is equal to 30^o, then the area of the parallelogram in sq. cm, is

CAT/2020.1(Quantitative Ability)

Question. 22

The area of the region satisfying the inequalities |x| - y ≤ 1, y ≥ 0, and y ≤ 1 is

CAT/2020.1(Quantitative Ability)

Question. 23

A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of the circle to the area of the rhombus is

CAT/2020.2(Quantitative Ability)

Question. 24

From the interior point of an equilateral triangle, perpendiculars are drawn on all three sides. The sum of the lengths of the perpendiculars is 's'. Then the area of the triangle is

CAT/2020.2(Quantitative Ability)

Question. 25

Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

CAT/2020.2(Quantitative Ability)

Question. 26

Let C1 and C2 be concentric circles such that the diameter of C1 is 2cm longer than that of C2. If a chord of C1 has length 6 cm and is a tangent to C2, then the diameter, in cm of C1 is

CAT/2020.3(Quantitative Ability)

Question. 27

The area, in sq. units, enclosed by the lines x = 2, y = |x - 2| + 4, the X-axis and the Y-axis is equal to

CAT/2020.3(Quantitative Ability)

Question. 28

The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is

CAT/2020.3(Quantitative Ability)

Question. 29

In a trapezium ABCD, AB is parallel to DC, BC is perpendicular to DC and ∠BAD = 45°. If DC = 5 cm, BC = 4 cm, the area of the trapezium in sq. cm is

CAT/2020.3(Quantitative Ability)

Question. 30

The points (2 , 1) and (-3 , -4) are opposite vertices of a parallelogram. If the other two vertices lie on the line x + 9y + c = 0, then c is

CAT/2019.1(Quantitative Ability)

Question. 31

With rectangular axes of coordinates, the number of paths from (1,1) to (8,10) via (4,6), where each step from any point (x,y) is either to (x,y+1) or to (x+1,y) is

CAT/2019.1(Quantitative Ability)

Question. 32

In a circle of radius 11 cm, CD is a diameter and AB is a chord of length 20.5 cm. If AB and CD intersect at a point E inside the circle and CE has length 7 cm, then the difference of the lengths of BE and AE, in cm, is

CAT/2019.1(Quantitative Ability)

Question. 33

Let T be the triangle formed by the straight line 3x + 5y - 45 = 0 and the coordinate axes. Let the circumcircle of T have radius of length L, measured in the same unit as the coordinate axes. Then, the integer closest to L is

CAT/2019.1(Quantitative Ability)

Question. 34

AB is a diameter of a circle of radius 5 cm. Let P and Q be two points on the circle so that the length of PB is 6 cm, and the length of AP is twice that of AQ. Then the length, in cm, of QB is nearest to

CAT/2019.1(Quantitative Ability)

Question. 35

Let S be the set of all points (x,y) in the x-y plane such that |x| + |y| ≤ 2 and |x| ≥ 1. Then, the area, in square units, of the region represented by S equals

CAT/2019.2(Quantitative Ability)

Question. 36

In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is

CAT/2019.2(Quantitative Ability)

Question. 37

Two circles, each of radius 4 cm, touch externally. Each of these two circles is touched externally by a third circle. If these three circles have a common tangent, then the radius of the third circle, in cm, is

CAT/2019.2(Quantitative Ability)

Question. 38

Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is 3/2 times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is

CAT/2019.2(Quantitative Ability)

Question. 39

Let ABC be a right-angled triangle with hypotenuse BC of length 20 cm. If AP is perpendicular on BC, then the maximum possible length of AP, in cm, is

CAT/2018.1(Quantitative Ability)

Question. 40

In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

CAT/2018.1(Quantitative Ability)

Question. 41

Given an equilateral triangle T1 with side 24 cm, a second triangle T2 is formed by joining the midpoints of the sides of T1. Then a third triangle T3 is formed by joining the midpoints of the sides of T2. If this process of forming triangles is continued, the sum of the areas, in sq cm, of infinitely many such triangles T1, T2, T3,... will be

CAT/2018.1(Quantitative Ability)

Question. 42

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

CAT/2018.1(Quantitative Ability)

Question. 43

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

CAT/2018.1(Quantitative Ability)

Question. 44

In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

CAT/2018.2(Quantitative Ability)

Question. 45

On a triangle ABC, a circle with diameter BC is drawn, intersecting AB and AC at points P and Q, respectively. If the lengths of AB, AC, and CP are 30 cm, 25 cm, and 20 cm respectively, then the length of BQ, in cm, is

CAT/2018.2(Quantitative Ability)

Question. 46

A chord of length 5 cm subtends an angle of 60° at the centre of a circle. The length, in cm, of a chord that subtends an angle of 120° at the centre of the same circle is

CAT/2018.2(Quantitative Ability)

Question. 47

A triangle ABC has area 32 sq units and its side BC, of length 8 units, lies on the line x = 4. Then the shortest possible distance between A and the point (0,0) is

CAT/2017.1(Quantitative Ability)

Question. 48

The area of the closed region bounded by the equation | x | + | y | = 2 in the two-dimensional plane is

CAT/2017.1(Quantitative Ability)

Question. 49

From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC. The area, in sq ft, of the remaining portion of triangle ABC is:

CAT/2017.1(Quantitative Ability)

Question. 50

Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC. Let BPC be an arc of a circle centered at A and lying between BC and BQC. If AB has length 6 cm then the area, in sq. cm, of the region enclosed by BPC and BQC is:

CAT/2017.1(Quantitative Ability)

Question. 51

Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is

CAT/2017.1(Quantitative Ability)

Question. 52

Let AB, CD, EF, GH, and JK be five diameters of a circle with center at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?

CAT/2017.1(Quantitative Ability)

Question. 53

The shortest distance of the point(1/2,1) from the curve y = |x - 1| + |x + 1| is

CAT/2017.2(Quantitative Ability)

Question. 54

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

CAT/2017.2(Quantitative Ability)

Question. 55

The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is

CAT/2017.2(Quantitative Ability)

Question. 56

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is

CAT/2017.2(Quantitative Ability)

Question. 57

ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees, then the value of ∠BCD (in degrees) is

CAT/2017.2(Quantitative Ability)

Question. 58

If three sides of a rectangular park have a total length 400 ft., then the area of the park is maximum when the length (in ft.) of its longer side is 

CAT/2017.2(Quantitative Ability)

Question. 59

Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB. If the perpendicular distance of P from each of AB, BC, and CA is 4(√2 - 1) cm, then the area, in sq. cm, of the triangle ABC is

CAT/2008(Quantitative Ability)

Question. 60

In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD = 3 cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC?

CAT/2008(Quantitative Ability)

Question. 61

Consider obtuse-angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer, then how many such triangles exist?

CAT/2008(Quantitative Ability)

Question. 62

Consider a square ABCD with midpoints E, F, G, H of AB, BC, CD and DA respectively. Let L denote the line passing through F and H. Consider points P and Q, on L and inside ABCD, such that the angles APD and BQC both equal 120°. What is the ratio of the area of ABQCDP to the remaining area inside ABCD?

CAT/2006(Quantitative Ability)

Question. 63

An equilateral triangle BPC is drawn inside a square ABCD. What is the value of the angle APD in degrees?

CAT/2005(Quantitative Ability)

Question. 64

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

CAT/2005(Quantitative Ability)

Question. 65

In the following figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that MN is perpendicular to AB. In addition, CG is perpendicular to AB such that AE : EB = 1 : 2, and DF is perpendicular to MN such that NL : LM = 1 : 2. The length of DH in cm is

CAT/2005(Quantitative Ability)

Question. 66

P, Q, S and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle. What is the perimeter of the quadrilateral PQSR?

CAT/2005(Quantitative Ability)

Question. 67

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

What is the ratio of the perimeter of the triangle ADC to that of the triangle BDC?

CAT/2005(Quantitative Ability)

Question. 68

Consider a triangle drawn on the X-Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

CAT/2004(Quantitative Ability)

Question. 69

A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son’s head are incident at the same point on the ground. If the heights of the lamp post , the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post, then how far (in meters) is the son standing from his father?

Comprehension

Directions for Questions: Answer the questions on the basis of the information given below.

In the adjoining figure, I and II are circles with centers P and Q respectively. The two circle touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the rartio 4 : 3. It is also known that the length of PO is 28 cm.

CAT/2004(Quantitative Ability)

Question. 70

What is the ratio of the length of PQ to that of QO?

Comprehension

Directions for Questions: Answer the questions on the basis of the information given below.

In the adjoining figure, I and II are circles with centers P and Q respectively. The two circle touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the rartio 4 : 3. It is also known that the length of PO is 28 cm.

CAT/2004(Quantitative Ability)

Question. 71

What is the radius of the circle II?

Comprehension

Directions for Questions: Answer the questions on the basis of the information given below.

In the adjoining figure, I and II are circles with centers P and Q respectively. The two circle touch each other and have a common tangent that touches them at points R and S respectively. This common tangent meets the line joining P and Q at O. The diameters of I and II are in the rartio 4 : 3. It is also known that the length of PO is 28 cm.

CAT/2004(Quantitative Ability)

Question. 72

The length of SO is

CAT/2004(Quantitative Ability)

Question. 73

In the adjoining figure, chord ED is parallel to the diameter AC of the circle. If ∠CBE = 65°, then what is the value of ∠DEC?

CAT/2004(Quantitative Ability)

Question. 74

On a semicircle with diameter AD, chord BC is parallel to the diameter. Further, each of the chords AB and CD has length 2, while AD has length 8. What is the length of BC?

CAT/2004(Quantitative Ability)

Question. 75

If the length of diagonals DF, AG and CE of the cube shown in the adjoining figure are equal to the three sides of a triangle, then the radius of the circle circumscribing that triangle will be

CAT/2004(Quantitative Ability)

Question. 76

A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?

CAT/2003(Quantitative Ability)

Question. 77

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

CAT/2003(Quantitative Ability)

Question. 78

In a triangle ABC, AB =6, BC = 8 and AC = 10. A perpendicular dropped from B, meets the side AC at D. A circle of radius BD (with centre B) is drawn. If the circle cuts AB and BC at P and Q respectively, then AP: QC is equal to

CAT/2003(Quantitative Ability)

Question. 79

In the diagram given below, ∠ABD = ∠CDB = ∠PQD = 90°. If AB : CD = 3: 1 the ratio of CD:PQ is

CAT/2003(Quantitative Ability)

Question. 80

In the figure below, the rectangle at the corner measures 10 cm × 20 cm. The corner A of the rectangle is also a point on the circumference of the circle . What is the radius of the circle in cm?

CAT/2003(Quantitative Ability)

Question. 81

A vertical tower OP stands at the centre O of a square ABCD. Let h and b denote the length of OP and AB respectively. Suppose ∠APB = 60° then the relationship between h and b can be expressed as

CAT/2003(Quantitative Ability)

Question. 82

In the figure given below, AB is the chord of a circle with centre O. AB is extended to C such that BC=OB. The straight line CO is produced to meet the circle at D. If ∠ACD = y degrees and ∠AOD = x degrees such that x=ky, then the value of k is

CAT/2003(Quantitative Ability)

Question. 83

In the figure (not drawn to scale) given below, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC and QD is parallel to CP. In ΔARC, ∠ARC = 90°, and in ΔPQS, ∠PSQ = 90°. The length of QS is 6 cms. What is ratio AP : PD?

CAT/2003(Quantitative Ability)

Question. 84

In the figure (not drawn to scale) given below, if AD = CD = BC, and ∠BCE = 96°, how much is ∠DBC?

CAT/2003(Quantitative Ability)

Question. 85

In the figure given below (not drawn to scale), A, B and C are three points on a circle with centre O. The chord BA is extended to a point T such that CT becomes a tangent to the circle at point C. If ∠ATC = 30° and ∠ACT = 50°, then the angle ∠BOA is

CAT/2002(Quantitative Ability)

Question. 86

In a triangle ABC, the internal bisector of the angle A meets BC at D. If AB = 4, AC = 3 and ∠A= 60° , then the length of AD is

CAT/2002(Quantitative Ability)

Question. 87

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

CAT/2002(Quantitative Ability)

Question. 88

In the figure, ACB is a right angled triangle. CD is the altitude. Circles are inscribed within the triangles ACD, BAD. P and Q are the centres of the circles. The distance PQ is

CAT/2002(Quantitative Ability)

Question. 89

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

CAT/2001(Quantitative Ability)

Question. 90

In triangle DEF shown below, points A, B, and C are taken on DE, DF and EF respectively such that EC = AC and CF = BC. If angle D = 40 degrees then what is angle ACB in degrees?

CAT/2001(Quantitative Ability)

Question. 91

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

CAT/2000(Quantitative Ability)

Question. 92

ABCD is a rhombus with diagonals AC and BD intersecting at the origin on the xy plane. If the equation of the line AD is x + y = 1 then the equation of line BC is

CAT/2000(Quantitative Ability)

Question. 93

There is a regular octagon A B C D E F G H, a frog is at the vertex A. It can jump on to any of the vertices except the exactly opposite vertex. The frog visits all the vertices exactly once and then reaches vertex E then how many times did it jump before reaching E?

CAT/2000(Quantitative Ability)

Question. 94

If the perimeter of a triangle is 14 and the sides are integers, then how many different triangles are possible?

CAT/2000(Quantitative Ability)

Question. 95

a, b and c are sides of a triangle . If a² + b² +c² = ab + bc + ac then the triangle will be

CAT/1999(Quantitative Ability)

Question. 96

CAT/1999(Quantitative Ability)

Question. 97

The figure below shows two concentric circles with centre O. PQRS is a square inscribed in the outer circle. It also circumscribes the inner circle, touching it at point B, C, D and A. What is the ratio of the perimeter of the outer circle to that of polygon ABCD?

CAT/1999(Quantitative Ability)

Question. 98

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

CAT/1998(Quantitative Ability)

Question. 99

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

CAT/1997(Quantitative Ability)

Question. 100

AB is the diameter of the given circle, while points C and D lie on the circumference as shown. If AB is 15 cm, AC is 12 cm and BD is 9 cm, find the area of the quadrilateral ACBD

CAT/1997(Quantitative Ability)

Question. 101

In the given figure, EADF is a rectangle and ABC is a triangle whose vertices lie on the sides of EADF.

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.

CAT/1996(Quantitative Ability)

Question. 102

In triangle ABC, angle B is a right angle. If (AC) is 6 cm, and D is the mid - point of side AC. The length of BD is

CAT/1996(Quantitative Ability)

Question. 103

The points of intersection of three lines , 2X + 3Y = 0,  5X - 7Y + 2 = 0, 9X - 5Y - 4 = 0

CAT/1996(Quantitative Ability)

Question. 104

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

CAT/1995(Quantitative Ability)

Question. 105

CAT/1995(Quantitative Ability)

Question. 106

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

CAT/1995(Quantitative Ability)

Question. 107

The length of a ladder is exactly equal to the height of the wall it is resting 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 tip of the wall. Then, the height of the wall is

CAT/1994(Quantitative Ability)

Question. 108

CAT/1994(Quantitative Ability)

Question. 109

Which one of the following cannot be the ratio of angles in a right angled triangle?