CAT QA Preparation 2025: Practice Geometry Questions and Solutions

1.a,b,c are integers that are the side of an obtuse-angled triangle. If ab = 4, find c.

2
3
1
More than one possible value

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

11
6
17
23

3.Sides of a triangle are 6, 10 and x respectively . For what value of x is the area of the the triangle the maximum?

8 cms
9 cms
12 cms
None of these

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

p:36v3
p:18v3
p:27v3
p:42v3

5.Perimeter of a ? having integer sides is equal to 15. Number of such triangles possible is?

7
6
8
5

6.A square is inscribed inside an equilateral triangle such that 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 an equilateral triangle?

12 : 12 + 7v3
24 : 24 + 7v3
18 : 12 + 15v3
6 : 6 + 5v3

7.?ABC has integer sides m,n,o such that mo = 12. How many such triangles are possible?

8
6
9
12

8.ABCDE is a regular pentagon. O is a point inside the pentagon such that AOB forms an equilateral triangle. What is the measure of angle OEA?

66°
48°
54°
72°

9. How many distinct triangles with integral valued sides and perimeter 14 can be formed?

6
5
4
3

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

Equilateral
Isosceles
Right-angled
Obtuse-angled

11.In the figure below, AB=BC=CD=DE=AE. O is a point in the regular pentagon such that AOB is an equilateral triangle. What is the angle measure of OEA?

 66^°
 56^°
 50°
 75°

12. Assume a right-angled triangle with an inradius 2 cm and a circumradius of 7 cm. What is the area of the triangle?

32 sq cms
31.5 sq cms
32.5 sq cms
33 sq cms

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

v3: 1
2: 1
2: v3
v2: 1

14.Find the altitude to side AC of triangle ABC having measurements AB = 20 cm, AC = 20 cm, BC = 30 cm.

10v7
8v7
7.5v7
15v7

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

6v3
4v3
8v3
2v3

16.A rectangular pool having dimensions 20 m width and 60 m in length is surrounded by a walkway of uniform width. If the total area of the walkway is 516m2, what is its width?

43 m
4.3m
3 m
3.5 m

17.The length of the common chord of two circles with radii 15 cm and 20 cm, having centres 25 cm apart, is:

18.Four horses are placed at the four corners of a square plot of side 14 m such that the adjacent horses can just reach one another. There is a small circular pond of area 20 m ² at the centre. Find the total ungrazed area.

22 m ²
42 m ²
84 m ²
168 m ²

19.Find the sum of the lengths of its diagonals of a rhombus having area 100 sq cms and perimeter 56cms.

33.40
34.40
31.20
32.30

20. Find the area of a Rhombus having perimeter of 12 and one angle = 120°

9 * (v3)/2
3 * (v3)/2
9 * v3
18 * v3

21.ABCD is a rectangle. The area of the isosceles right triangle ABE is:
7 sq. cm; EC = 3(BE). The area of the rectangle ABCD (in cm²) is:

21 cm²
28 cm²
42 cm²
56 cm²

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

144 cm²
121 cm²
156 cm²
168 cm²

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

 π:36√3
 π:18√3
 π:27√3
None of these

24.A rectangular pool is 20 m wide and 60 m long. It is surrounded by a walkway of uniform width. The total area of the walkway is 516m2, find its width.

43 m
4.3m
3 m
3.5 m

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

29 cm
13 cm
26 cm
Cannot be determined due to lack of information