# CAT Practice : Geometry - Triangles

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The following topics are covered in the CAT quant section from Geometry - Triangles. Detailed explanatory answers, solution videos and slide decks are also provided.
1. ### Triangles - Integer values

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. ### Triangles - Integer values

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. ### 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
None of these
• Maximum possible area
• Hard
4. ### 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$\sqrt {3}$
2. π : 18$\sqrt {3}$
3. π : 27$\sqrt {3}$
4. π : 42$\sqrt {3}$
π : 27$\sqrt {3}$
• Area Properties
• Hard
5. ### Number of triangles

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. ### 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$\sqrt {3}$
2. 24 : 24 + 7$\sqrt {3}$
3. 18 : 12 + 15$\sqrt {3}$
4. 6 : 6 + 5$\sqrt {3}$
12 : 12 + 7$\sqrt {3}$
• Symmetric figures
• Hard
7. ### Triangles - Integer values

△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. ### Regular pentagon

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. ### Triangle 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
Acute-angled
• Acute or Obtuse
• Medium
10. ### 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
32 sq cms
• Medium
11. ### Diagonals of octagon

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

1. ${\surd }$3 : 1
2. 2 : 1
3. 2 : ${\surd }$3
4. ${\surd }$2 : 1
12. ### Altitude of triangle

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

1. 10${\surd }$7
2. 8${\surd }$7
3. 7.5${\surd }$7
4. 15${\surd }$7
13. ### Hexagon inscribed inside circle

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(3${\Pi }$ − (9${\surd }$3)/2)
2. 1/6(2${\Pi }$ − (6${\surd }$3)/2)
3. 1/6(3${\Pi }$ − (8${\surd }$3)/2)
4. 1/6(6${\Pi }$ − (15${\surd }$3)/2)
• Hexagon inscribed inside circle
• Hard
14. ### Chords of a circle

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${\surd }$3
2. 4${\surd }$3
3. 8${\surd }$3
4. 2${\surd }$3
15. ### Area of the Triangle

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${\surd }$3
2. ${\surd }$3
3. ${\frac{√3}{2} }$
4. 3
16. ### Rhombus - Length of the diagonals

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 - Length of the diagonals
• Medium
17. ### Rhombus - Area

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

1. 9*${\frac{√3}{2} }$
2. 3*${\frac{√3}{2} }$
3. 9*${\surd }$3
4. 18*${\surd }$3
18. ### Area of an isosceles triangle

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
• Area of an isosceles triangle
• Medium

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

1. $26^{\frac{1}{2}}$
2. $13^{\frac{1}{2}}$
3. $24^{\frac{1}{2}}$
4. $52^{\frac{1}{2}}$
20. ### Triangle

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°
21. ### Area of Rhombus

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 $\sqrt{2}$cm2
2. 9 $\sqrt{2}$cm2
3. 15 $\sqrt{2}$cm2
4. 12 $\sqrt{2}$cm2
22. ### 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 $\sqrt{3}$ : 16
2. 3 $\sqrt{3}$ : 4
3. 9 $\sqrt{3}$ : 4
4. 3 $\sqrt{3}$ : 16
• Circle, Square and Triangle
• Medium
23. ### Isosceles Triangle

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

1. $\sqrt{231}$ units
2. 12$\sqrt{66}$ units
3. 24 units
4. 5 $\sqrt{231}$ units
24. ### Ratio of Areas

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. ${ \Pi : (6 + 4\surd 3)}$
2. ${3\Pi : (6 + 4\surd 3)}$
3. ${2\Pi : (6 + 4\surd 3)}$
4. ${ \Pi : (6 + 2\surd 3)}$
25. ### Ratio of Areas

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. ${ \surd 3 : (5 + 4\surd 3)}$
2. ${2\surd 3 : (7 + 4\surd 3)}$
3. ${4\surd 3 : (7 + 4\surd 3)}$
4. ${ 4\surd 3 : (5 + 2\surd 3)}$
26. ### Ratio of Areas

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

1. 2 : ${\Pi}$ , 4 : ${\Pi}$
2. 4 : ${\Pi}$ , 2 : ${\Pi}$
3. 1 : ${\Pi}$ , 4 : ${\Pi}$
4. 2 : ${\Pi}$ , 1 : ${\Pi}$
27. ### Ratio of Areas

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${\surd 3}$ : ${\Pi}$ , 3${\surd 3}$ : ${16\Pi}$
2. 3${\surd 3}$ : ${4\Pi}$, 3${\surd 3}$ : ${\Pi}$
3. ${\surd 3}$ : ${\Pi}$ , 3${\surd 3}$ : ${4\Pi}$
4. ${\surd 3}$ : ${\Pi}$ , ${\surd 3}$ : ${16\Pi}$
28. ### Ratio of Areas

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${\surd 3}$ : ${3\Pi}$ , 3${\surd 3}$ : ${4\Pi}$
2. 3${\surd 3}$ : ${\Pi}$ , 3${\surd 3}$ : ${4\Pi}$
3. 3${\surd 3}$ : ${2\Pi}$, 2${\surd 3}$ : ${\Pi}$
4. ${\surd 3}$ : ${\Pi}$ , ${\surd 3}$ : ${4\Pi}$
29. ### Distance between circumcenter and orthocenter

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. ${\surd 30}$ cm
• Distance - ortho and circumcenter
• Easy
30. ### Circles touching externally

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 + ${\surd 7}$)/2 cm2
3. 27/2 cm2
4. (3${\surd 41}$)/2 cm2
• Circles touching externally
• Medium
31. ### Circumference of a circle

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 ${\surd 34}$ cm
3. 34π/3 cm
4. π ${\surd 31}$/3 cm
• Circumference of a circle
• Medium
32. ### Data Sufficiency

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. ### Side of a triangle

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. ${\surd 23}$ cm
II. ${\surd 73}$ cm
III. (4.5 - ${\surd 3.25}$) cm
IV. (4 + ${\surd 33}$) cm
V. (9 + ${\surd 13}$) cm

1. Only II and IV
2. Only I and III
3. Only I, II and V
4. Only II, III and IV
34. ### Number of parallelograms

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. ### Ratio of circumferences

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 : ${\surd 2}$
4. π : ${\surd 2}$

If PQ∥RS, find the value of x

1. 7
2. 3
3. Both A & B
4. None of these
37. ### Inscribed Circles

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

The radius of the circle is:-

1. ${\frac{\surd {2}+ 1}{2}}$
2. ${\surd {2} - \frac{1}{2}}$
3. ${1 - \surd 2}$
4. ${\surd {2} - 1}$
38. ### 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. ### Circles

In the above figure which of the following holds good?

1. ∠SQO = ∠ROP
2. 2 ∠ROP = ∠SOR
3. ∠POR = ∠ASO
4. ∠QOX’ = ∠SOR + ∠ROP
40. ### Basic Geometry

In the below figure:-

If ${\frac{PQ}{PR} = \frac{PR}{RQ}}$ , then

1. ${\frac{PR}{QR}} > 2$
2. ${\frac{PR}{QR}} = 2$
3. ${\frac{PR}{QR}} < 2$
4. can’t be determined
41. ### Basic Geometry

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, ${\frac{QT}{TR}} > 2$ takes the value.

1. $\frac{QT}{TR} < \frac{1}{3}$
2. $\frac{1}{3} < \frac{QT}{TR} < 1$
3. $\frac{QT}{TR} > 1$
4. can’t be determined
42. ### Basic Geometry

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. ### Basic Geometry

In the above 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
2. 5,000
3. 8,750
4. 12,500
44. ### Basic Geometry

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 above. 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
2. ${\frac{5}{2}}$ m
3. ${\frac{5}{8}}$ m
4. ${\frac{15}{8}}$ m
45. ### Basic Geometry

The number of sides in a regular polygon is ‘T’ times the number of diagonals in it. What is the interior angle of this polygon in terms of T?

1. ${180 * \frac{(T+2)}{(3T+2)}}$
2. ${540 * \frac{(T+2)}{(3T+2)}}$
3. ${360 * \frac{(T+2)}{(3T+2)}}$
4. ${90 * \frac{(T+2)}{(3T+2)}}$

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