# CAT Questions | CAT Geometry Questions

###### CAT Quantitative Aptitude | CAT Geometry Triangles , Circles

A CAT Geometry question that appears in the Quantitative Aptitude section of the CAT Exam broadly tests an aspirant on the concepts - Triangles, Circles, Quadrliaterals, Polygons & mixture of the above mentioned concepts. In CAT Exam, one can generally expect to get 4~6 questions from CAT Geometry. CAT Geometry is an important topic with lots of weightage in the CAT Exam. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

1. #### CAT Geometry: Obtuse Triangles - Integers

x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z.

1. 2
2. 3
3. 1
4. More than one possible value of z exists

2. #### CAT Geometry: Isosceles Triangles - Integers

How many isosceles triangles with integer sides are possible such that sum of two of the side is 12?

1. 11
2. 6
3. 17
4. 23

3. #### CAT Geometry: Triangles - Area

Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?

1. 8 cms
2. 9 cms
3. 12 cms
4. None of these

4. #### CAT Geometry: Equilateral triangle and circle

Two circles are placed in an equilateral triangle as shown in the figure. What is the ratio of the area of the smaller circle to that of the equilateral triangle

1. π:36√3
2. π:18√3
3. π:27√3
4. π:42√3

5. #### CAT Geometry: Triangles - Perimeter

Perimeter of a △ with integer sides is equal to 15. How many such triangles are possible?

1. 7
2. 6
3. 8
4. 5

6. #### CAT Geometry: Equilateral Triangle and Square

There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?

1. 12 : 12 + 7√3
2. 24 : 24 + 7√3
3. 18 : 12 + 15√3
4. 6 : 6 + 5√3

7. #### CAT Geometry: Triangles - Integers

△ABC has integer sides x, y, z such that xz = 12. How many such triangles are possible?

1. 8
2. 6
3. 9
4. 12

8. #### CAT Geometry: Regular Polygon

ABCDE is a regular pentagon. O is a point inside the pentagon such that AOB is an equilateral triangle. What is ∠OEA?

1. 66°
2. 48°
3. 54°
4. 72°

9. #### CAT Geometry: Triangles Properties

△ has sides a2, b2 and c2. Then the triangle with sides a, b, c has to be:

1. Right-angled
2. Acute-angled
3. Obtuse-angled
4. Can be any of these three

10. #### CAT Geometry: Right Triangle Properties

Consider a right–angled triangle with inradius 2 cm and circumradius of 7 cm. What is the area of the triangle?

1. 32 sq cms
2. 31.5 sq cms
3. 32.5 sq cms
4. 33 sq cms

11. #### CAT Geometry: Regular Polygon

What is the ratio of longest diagonal to the shortest diagonal in a regular octagon?

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

12. #### CAT Geometry: Altitude of a Triangle

Find the altitude to side AC of triangle with side AB = 20 cm, AC = 20 cm, BC = 30 cm.

1. 10√7
2. 8√7
3. 7.5√7
4. 15√7

13. #### CAT Geometry: Regular Polygon

ABCDEF is a regular hexagon inscribed inside a circle. If the shortest diagonal of the hexagon is of length 3 units, what is the area of the shaded region.

1. 1/6(3π − (9√3)/2)
2. 1/6(2π − (6√3)/2)
3. 1/6(3π − (8√3)/2)
4. 1/6(6π − (15√3)/2)

14. #### CAT Geometry: Circles

A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = 1/3 of RT. Find minimum value of PQ.

1. 6√3
2. 4√3
3. 8√3
4. 2√3

15. #### CAT Geometry: Triangles - Area

Triangle has perimeter of 6 + 2√3 . One of the angles in the triangle is equal to the exterior angle of a regular hexagon another angle is equal to the exterior angle of a regular 12-sided polygon. Find area of the triangle.

1. 2√3
2. √3
3. √3/2
4. 3

Area of a Rhombus of perimeter 56 cms is 100 sq cms. Find the sum of the lengths of its diagonals

1. 33.40
2. 34.40
3. 31.20
4. 32.30

Rhombus has a perimeter of 12 and one angle = 120°. Find its area.

1. 9 * (√3)/2
2. 3 * (√3)/2
3. 9 * √3
4. 18 * √3

18. #### CAT Geometry: Triangles - Area

Circle with center O and radius 25 cms has a chord AB of length of 14 cms in it. Find the area of triangle AOB

1. 144 cm2
2. 121 cm2
3. 156 cm2
4. 168 cm2

19. #### CAT Geometry: Circles

Two mutually perpendicular chords AB and CD intersect at P. AP = 4, PB = 6, CP = 3. Find radius of the circle.

1. 31.25(1/2)
2. 37.5(1/2)
3. 26(1/2)
4. 52(1/2)

20. #### CAT Geometry: Triangles

Triangle ABC has angles A = 60° and B = 70°. The incenter of this triangle is at I. Find angle BIC.

1. 90°
2. 130°
3. 80°
4. 120°

Rhombus of side 6 cm has an angle equal to the external angle of a regular octagon. Find the area of the rhombus.

1. 18√2 cm2
2. 9√2 cm2
3. 15√2 cm2
4. 12√2 cm2

22. #### CAT Geometry: Circle, Square and Triangle

A circle inscribed in a square of side 2 has an equilateral triangle inscribed inside it. What is the ratio of areas of the equilateral triangle to that of the square?

1. 9√3 : 16
2. 3√3 : 4
3. 9√3 : 4
4. 3√3 : 16

23. #### CAT Geometry: Isosceles Triangles

An acute-angled isosceles triangle has two of its sides equal to 10 and 16. Find the area of this triangle.

1. √231 units
2. 12√66 units
3. 24 units
4. 5√231 units

24. #### CAT Geometry: Triangles - Area

Three equal circles are placed inside an equilateral triangle such that any circle is tangential to two sides of the equilateral triangle and to two other circles. What is the ratio of the areas of one circle to that of the triangle?

1. π : (6+4√3)
2. 3π : (6+4√3)
3. 2π : (6+4√3)
4. π : (6+2√3)

25. #### CAT Geometry: Triangle and Square

There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?

1. √3 : (5 + 4√3)
2. 2√3 : (7 + 4√3)
3. 4√3 : (7 + 4√3)
4. 4√3 : (5 + 2√3)

26. #### CAT Geometry: Squares and Circles

Consider Square S inscribed in circle C, what is the ratio of the areas of S and C? And, Consider Circle Q inscribed in Square S, what is the ratio of the areas of S and Q?

1. 2:π, 4:π
2. 4:π, 2:π
3. 1:π, 4:π
4. 2:π, 1:π

27. #### CAT Geometry: Triangles - Area

Consider equilateral triangle T inscribed in circle C, what is ratio of the areas of T and C? Consider Circle C inscribed in equilateral triangle T, what is ratio of the areas of T and C?

1. 3√3:π , 3√3:16π
2. 3√3:4π , 3√3:π
3. √3:π , 3√3:4π
4. √3:π , √3:16π

28. #### CAT Geometry: Regular Octagon

Consider Regular Hexagon H inscribed in circle C, what is ratio of the areas of H and C? Consider Circle C inscribed in Regular Hexagon H, what is ratio of the areas of H and C?

1. 2√3 : 3π , 3√3 : 4π
2. 3√3 : π , 3√3 : 4π
3. 3√3 : 2π, 2√3 : π
4. √3 : π , √3 : 4π

29. #### CAT Geometry: Triangles

What is the distance between the orthocentre and the circumcenter of a triangle who sides measure 24 cm, 26 cm and 10 cm?

1. 13 cm
2. 12 cm
3. 7.5 cm
4. √30 cm

30. #### CAT Geometry: Circles and Triangles

Two circles with centres O1 and O2 touch each other externally at a point R. AB is a tangent to both the circles passing through R. P’Q’ is another tangent to the circles touching them at P and Q respectively and also cutting AB at S. PQ measures 6 cm and the point S is at distance of 5 cms and 4 cms from the centres of the circles. What is the area of the triangle SO1O2?

1. 9 cm2
2. 3(4+√7)/2 cm2
3. 27/2 cm2
4. (3√41)/2 cm2

31. #### CAT Geometry: Circles

What is the circumference of the below circle given that AB is the diameter and XY is perpendicular to AB?

1. 8π cm
2. π√34 cm
3. 34π/3 cm
4. π√31/3 cm

32. #### CAT Geometry: Circles

Find ∠PRB. Given
I. ∠BPQ = 22 and O is the centre of the circle
II. ∠RBP = 54 and chord PQ is parallel to AB

1. Either I or II individually is sufficient
2. Both I and II together are required
3. One of the statements alone is sufficient
4. Need more data

33. #### CAT Geometry: Triangles

The two sides of a triangle are 8 cm and 9 cm and one angle is 60. Which of the following can be the length of its third side?
I. √23 cm
II. √73 cm
III.(4.5 - √3.25) cm
IV. (4 + √33) cm
V. (9 + √13) cm

1. Only II and IV
2. Only I and III
3. Only I, II and V
4. Only II, III and IV

There is a set of parallel lines with x lines in it and another set of parallel lines with y lines in it. The lines intersect at 12 points. If x > y, find the maximum number of parallelograms that can be formed.

1. 16
2. 15
3. 18
4. 33

35. #### CAT Geometry: Polygon and Circles

There are 2 concentric circles, one big and one small. A square ABCD is inscribed inside the big circle while the same square circumscribes the small circle. The square touches the small circle at points P, Q, R and S. Determine the ratio of circumference of big circle to the polygon PQRS.

1. π : 2
2. 2 : π
3. 2 : √2
4. π : √2

If PQ || RS, find the value of x

1. 7
2. 3
3. Both A & B
4. None of these

37. #### CAT Geometry: Circles

A circle is inscribed in a semi-circle as shown:-

The radius of the circle possibly is:-

1. (√2 + 1)/2
2. √2 − 1/2
3. 1 − √2
4. √2 − 1

38. #### CAT Geometry: Circles

Two circles of radius 5 cm have a direct tangent PQ and an indirect tangent RS. Find the length of PQ if RS = 24 cm.

1. 29 cm
2. 13 cm
3. 26 cm
4. Cannot be determined due to lack of information

39. #### CAT Geometry: Circles and Triangles

In the below figure which of the following holds good?

1. ∠SQO = ∠ROP
2. 2 ∠ROP = ∠SOR
3. ∠POR = ∠ASO
4. ∠QOX’ = ∠SOR + ∠ROP

In the below figure, If $$frac{PR}{PQ}\\$ = $\frac{PQ}{RQ}\\$, then 1. $\frac{PQ}{QR}\\$ > 2 2. $\frac{PQ}{QR}\\$ = 2 3. $\frac{PQ}{QR}\\$ < 2 4. can’t be determined 41. #### CAT Geometry: Triangles A right angled triangle PQR is such that ∠PRQ = 90° and QR = 4 cm T is a point on QR such that PT = 3 cm, and perimeter of triangle PQT = Perimeter of triangle PTR Then, QT/TR takes the value: 1. QT/TR < 1/3 2. 1/3 < QT/TR < 1 3. QT/TR > 1 4. can’t be determined 42. #### CAT Geometry: Triangles M and N are two points on the side PQ and PR of a triangle PQR respectively such that MNQR is a trapezium and MN:QR = 2:5. Find the ratio of the area of triangle PMN : Trapezium MNQR. 1. 4:25 2. 1:2 3. 4:21 4. 4:10 43. #### CAT Geometry: Triangles In the below figure, ΔABC is right angled and AC = 100 cm. Also, AD = DE = EF = FC. Find the value of: BD2 + BE2 + BF2$in cm2)

1. 10,000 cm2
2. 5,000 cm2
3. 8,750 cm2
4. 12,500 cm2

44. #### CAT Geometry: Triangles

The Olympics committee came up with a new rule. The flag of the gold medal winning team would be hoisted to the right (AB) at 5m. The flag of silver medal winning team would be hoisted to the left (PQ) at a height of 3m. The flag (MN) of bronze medal winning team would be hoisted at the point of intersection of the line joining the top of each of AB and PQ to the foot of other, as shown in the figure below. A and P are 8m apart. In a wrestling event, India won the bronze medal. Find the height at which the Indian flag was hoisted.

1. 2 m

1. #### CAT 2018 Question Paper Slot 2 - Geometry

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

1. 5√3
2. 6√2
3. 8

2. #### CAT 2018 Question Paper Slot 1 - Geometry

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

1. #### CAT 2017 Question Paper Slot 2 - Geometry

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 [TITA]

1. #### CAT 2017 Question Paper Slot 1 - Geometry

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:

1. 225√3
2. $$frac{500}{√3}\\$ 3. $\frac{275}{√3}\\$ 4. $\frac{250}{√3}\\$ 2. #### CAT 2017 Question Paper Slot 1 - Geometry 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: 1. 9π - 18 2. 18 3. 9 3. #### CAT 2017 Question Paper Slot 1 - Geometry 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:$TITA)

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

1. #### XAT 2020 Question Paper - QADI

Two lighthouses, located at points A and B on the earth, are 60 feet and 40 feet tall respectively. Each lighthouse is perfectly vertical and the land connecting A and B is perfectly flat. The topmost point of the lighthouse at A is A’ and of the lighthouse at B is B’. Draw line segments A’B and B’A, and let them intersect at point C’. Drop a perpendicular from C’ to touch the earth at point C. What is the length of CC’ in feet?

1. 20
2. 30
3. 24
4. The distance between A and B is also needed to solve this
5. 25

2. #### XAT 2020 Question Paper - QADI

A rectangular field is 40 meters long and 30 meters wide. Draw diagonals on this field and then draw circles of radius 1.25 meters, with centers only on the diagonals. Each circle must fall completely within the field. Any two circles can touch each other but should not overlap.
What is the maximum number of such circles that can be drawn in the field?

1. 39
2. 40
3. 37
4. 36
5. 38

3. #### XAT 2020 Question Paper - QADI

XYZ is an equilateral triangle, inscribed in a circle. P is a point on the arc YZ such that X and P are on opposite sides of the chord YZ. Which of the following MUST always be true?

1. XZ + YP = XY + PZ
2. XP = YP + PZ
3. XP + PZ = XY + YP
4. XP = XY
5. XP = XY + YZ

4. #### XAT 2019 Question Paper - QADI

In the picture below, EFGH, ABCD are squares, and ABE, BCF, CDG, DAH are equilateral triangles. What is the ratio of the area of the square EFGH to that of ABCD?

1. √3+2
2. 3+√2
3. 1+√3
4. √2+2
5. √2+√3

5. #### XAT 2019 Question Paper - QADI

What is the maximum number of points that can be placed on a circular disk of radius 1 metre (some of the points could be placed on the bounding circle of the disk) such that no two points are at a distance of less than 1 metre from each other?

1. 6
2. 9
3. 7
4. 8
5. 5

6. #### XAT 2019 Question Paper - QADI

The figure below shows two right angled triangles ∆OAB and ∆OQP with right angles at vertex A and P, respectively, having the common vertex O, The lengths of some of the sides are indicated in the figure. (Note that the figure is not drawn to scale.) AB and OP are parallel. What is ∠QOB?

1. $$tan ^{-1}$2 / 3$)
2. 45°
3. 30°
4. $$tan ^{-1}$3 / 2$)
5. 60°

7. #### XAT 2019 Question Paper - QADI

Let ABC be an isosceles triangle. Suppose that the sides AB and AC are equal and let the length of AB be x cm. Let b denote the angle ∠ABC and sin b = 3/5. If the area of the triangle ABC is M square cm, then which of the following is true about M?

1. M < $$frac{x^{2}}{4}$ 2. M ≥ x2 3. $\frac{3 x^{2}}{4}$ ≤ M < x2 4. $\frac{x^{2}}{2}$ ≤M < $\frac{3 x^{2}}{4}$ 5. $\frac{x^{2}}{4}$ ≤ M < $\frac{x^{2}}{2}$ 8. #### XAT 2018 Question Paper - QADI If the diagonals of a rhombus of side 15 cm are in the ratio 3:4, find the area of the rhombus. 1. 54 sq. cm. 2. 108 sq. cm. 3. 144 sq. cm. 4. 200 sq. cm. 5. None of the above 9. #### XAT 2018 Question Paper - QADI Two circles with radius 2R and √2R intersect each other at points A and B. The centers of both the circles are on the same side of AB. O is the center of the bigger circle and ∠AOB is 60°. Find the area of the common region between two circles. 1.$√3 – π – 1)R2
2. (√3 – π)R2
3. (13π/6 + 1 - √3)R2
4. (13π/6 + √3)R2
5. None of the above

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

1. #### IPMAT 2020 Sample Paper - IPM Rohtak Quants

A line from center to circumference of a circle is known as

1. diameter
3. area
4. midpoint

2. #### IPMAT 2020 Question Paper - IPM Indore Quants

The number of acute angled triangles whose sides are three consecutive positive integers and whose perimeter is at most 100 is

1. 28
2. 29
3. 31
4. 33

5. #### IPMAT 2019 Question Paper - IPM Indore Quants

A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

1. (2 + √3)
2. (1 + √2)
3. (7 + 4√3)
4. (97 + 56√3)

6. #### IPMAT 2019 Question Paper - IPM Indore Quants

Points P, Q, R and S are taken on sides AB, BC, CD and DA of square ABCD respectively, so thatAP : PB = BQ : QC = CR : RD = DS : SA = 1 : n . Then the ratio of the area of PQRS to the area of ABCD is

1. 1 : (1 + n)
2. 1 : n
3. 1 + n2 : (1 + n)2
4. (1 + n ) : (1 + n2)

7. #### IPMAT 2019 Question Paper - IPM Indore Quants

On a circular path of radius 6 m a boy starts from a point A on the circumference and walks along a chord AB of length 3 m. He then walks along another chord BC of length 2 m to reach point C. The point B lies on the minor arc AC. The distance between point C from point A is

1. $$frac{\sqrt{15} + \sqrt{35}}{2}\\$ m 2. 8 m 3. √13 m 4. 6 m 8. #### IPMAT 2019 Question Paper - IPM Indore Quants Two points on a ground are 1 m apart. If a cow moves in the field in such a way that it's distance from the two points is always in ratio 3: 2 then 1. the cow moves in a straight line 2. the cow moves in a circle 3. the cow moves in a parabola 4. the cow moves in a hyperbola ###### Prepare for CAT 2022 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2021Enroll at 49,000/- 44,000/- Online Classroom Batches Starting Now! ###### Best CAT Coaching in Chennai Introductory offer of 5000/- Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ## CAT Preparation Online | CAT Geometry questions Videos On YouTube #### Other useful sources for Geometry Question | Geometry Triangles Circles Quadrilaterals Sample Questions ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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