Geometry Aptitude Practice Test Questions -- Free Practice

Question 1

In a triangle ABC, AB = 3 cm, AC = 4 cm and the length of the median AD to side BC is 2.5 cm. Then the side BC is equal to:

A

1✓✗

B

3.25✓✗

C

5✓✗

D

Cannot be determined✓✗

Explanation

By the median length formula, AD² = 2AB² + 2AC² - BC²4.
4 × 6.25 = 18 + 32 - BC², so BC² = 50 - 25 = 25 and BC = 5 cm.

Question 2

If in a right angle triangle, the three sides are 3, 4, 5, what is the circumradius of the triangle?

A

5✓✗

B

2✓✗

C

9✓✗

D

2.5✓✗

Explanation

For a right triangle, the circumradius equals half the hypotenuse: 52 = 2.5.

Question 3

Each interior angle of a regular polygon is 120 degrees greater than each exterior angle. How many sides does the polygon have?

A

6✓✗

B

8✓✗

C

12✓✗

D

3✓✗

Explanation

Interior+Exterior=180 and Interior=Exterior+120 gives Exterior=30 degrees, so the number of sides = 36030 = 12.

Question 4

The difference between the obtuse angle and one of the acute angles of a triangle is 20 degrees. The difference between the two acute angles of the triangle is 56 degrees. Which of the following is one of the two acute angles?

A

12 degrees✓✗

B

16 degrees✓✗

C

22 degrees✓✗

D

18 degrees✓✗

Explanation

With O=a1+20 and a1-a2=56, and O+a1+a2=180, solving gives a1=72 and a2=16.

Question 5

If the length of the smallest altitude of a triangle is 6 cm, then the inradius is always:

A

>3 cm✓✗

B

3 cm✓✗

C

<3 cm✓✗

D

Any of these✓✗

Explanation

It is a known geometric result that the inradius of a triangle is always less than half its smallest altitude, i.e. less than 3 cm here.

Question 6

For a circle with radius = 5 cm, what is the length of the chord which is at a distance of 3 cm from the centre?

A

4 cm✓✗

B

6 cm✓✗

C

7 cm✓✗

D

8 cm✓✗

Explanation

Half-chord = √5²-3²=4, so the full chord = 8 cm.

Question 7

The centres of 3 circles, which touch each other externally, form a triangle of sides 14 cm, 18 cm and 20 cm. Find the radius of the smallest circle.

A

6 cm✓✗

B

7 cm✓✗

C

8 cm✓✗

D

12 cm✓✗

Explanation

With r1+r2=14, r2+r3=18, r1+r3=20, adding gives r1+r2+r3=26, so r3=12, r1=8, r2=6. The smallest radius is 6 cm.

Question 8

Two angles are supplementary, and one of them is 36° more than the other. What is the larger angle?

A

108°✓✗

B

72°✓✗

C

126°✓✗

D

116°✓✗

Explanation

Supplementary angles add up to 180°. Let the angles be x and x + 36.
x + x + 36 = 180, so 2x = 144 and x = 72°.
The larger angle = 72 + 36 = 108°.

Question 9

The supplement of an angle is four times its complement. What is the angle?

A

30°✓✗

B

45°✓✗

C

60°✓✗

D

75°✓✗

Explanation

Let the angle be x. Supplement = 180 - x and complement = 90 - x.
180 - x = 4(90 - x) = 360 - 4x, so 3x = 180 and x = 60°.
Check: supplement 120° = 4 × complement 30°.

Question 10

AB and CD are parallel lines, with A and C on the left and B and D on the right. A point E lies between the two lines, on the same side of BD as A and C. If ∠ABE = 40° and ∠CDE = 35°, what is ∠BED?

A

105°✓✗

B

75°✓✗

C

140°✓✗

D

145°✓✗

Explanation

Draw a line through E parallel to AB (and so parallel to CD).
It splits ∠BED into two parts. The part next to B equals ∠ABE = 40° (alternate angles), and the part next to D equals ∠CDE = 35° (alternate angles).
∠BED = 40 + 35 = 75°.

Question 11

An exterior angle of a triangle is 110°, and one of the two interior opposite angles is 45°. What is the other interior opposite angle?

A

55°✓✗

B

65°✓✗

C

70°✓✗

D

25°✓✗

Explanation

An exterior angle of a triangle equals the sum of the two interior opposite angles.
Other angle = 110 - 45 = 65°.

Question 12

Two sides of a triangle are 7 cm and 11 cm. How many different whole-number lengths (in cm) are possible for the third side?

A

11✓✗

B

12✓✗

C

13✓✗

D

14✓✗

Explanation

The third side must be greater than the difference and less than the sum of the other two sides: 11 - 7 < x < 11 + 7, so 4 < x < 18.
Whole numbers from 5 to 17: 17 - 5 + 1 = 13 values.

Question 13

In triangle ABC, a line DE is drawn parallel to BC, with D on AB and E on AC. If AD = 4 cm and DB = 6 cm, what is the ratio of the area of triangle ADE to the area of quadrilateral DBCE?

A

4 : 21✓✗

B

4 : 25✓✗

C

2 : 3✓✗

D

16 : 25✓✗

Explanation

Since DE is parallel to BC, triangle ADE is similar to triangle ABC, with ratio of sides AD : AB = 4 : 10 = 2 : 5.
Ratio of areas = 2² : 5² = 4 : 25.
So if ADE = 4 parts, ABC = 25 parts and DBCE = 25 - 4 = 21 parts.
ADE : DBCE = 4 : 21.

Question 14

In triangle ABC, ∠A = 90° and AD is the perpendicular from A to BC. If BD = 4 cm and DC = 9 cm, what is the length of AB?

A

6 cm✓✗

B

3√13 cm✓✗

C

2√10 cm✓✗

D

2√13 cm✓✗

Explanation

The perpendicular from the right angle to the hypotenuse splits the triangle into two triangles similar to ABC.
From triangle ABD similar to triangle CBA: AB² = BD × BC = 4 × (4 + 9) = 52.
AB = √52 = 2√13 cm.
(Also AD² = BD × DC = 36, so AD = 6 cm, and AC² = 9 × 13 = 117, so AC = 3√13 cm.)

Question 15

In parallelogram ABCD, ∠A is 30° less than twice ∠B. What is ∠A?

A

70°✓✗

B

100°✓✗

C

110°✓✗

D

120°✓✗

Explanation

Adjacent angles of a parallelogram add up to 180°: A + B = 180.
A = 2B - 30, so 3B - 30 = 180, B = 70° and A = 110°.

Question 16

The diagonals of a rhombus are 16 cm and 30 cm. What is the perimeter of the rhombus?

A

46 cm✓✗

B

68 cm✓✗

C

92 cm✓✗

D

34 cm✓✗

Explanation

The diagonals of a rhombus bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 8 cm and 15 cm.
Side = √(64 + 225) = √289 = 17 cm.
Perimeter = 4 × 17 = 68 cm.

Question 17

ABCD is a cyclic quadrilateral in which ∠A = (2x + 10)° and ∠C = (3x - 5)°. What is ∠A?

A

100°✓✗

B

70°✓✗

C

35°✓✗

D

80°✓✗

Explanation

Opposite angles of a cyclic quadrilateral add up to 180°.
(2x + 10) + (3x - 5) = 180, so 5x = 175 and x = 35.
∠A = 2 × 35 + 10 = 80° (and ∠C = 100°).

Question 18

In a quadrilateral ABCD, the bisectors of ∠A and ∠B meet at O. If ∠C = 100° and ∠D = 60°, what is ∠AOB?

A

80°✓✗

B

100°✓✗

C

160°✓✗

D

90°✓✗

Explanation

Angles of a quadrilateral add up to 360°, so A + B = 360 - 100 - 60 = 200°.
In triangle AOB, ∠OAB + ∠OBA = A + B2 = 100°.
∠AOB = 180 - 100 = 80°. (In general, ∠AOB = C + D2.)

Question 19

What is the sum of the interior angles of a polygon with 9 sides?

A

1080°✓✗

B

1260°✓✗

C

1440°✓✗

D

1620°✓✗

Explanation

Sum of interior angles = (n - 2) × 180° = 7 × 180 = 1260°.

Question 20

Each interior angle of a regular polygon measures 156°. How many sides does the polygon have?

A

12✓✗

B

18✓✗

C

15✓✗

D

20✓✗

Explanation

Each exterior angle = 180 - 156 = 24°.
The exterior angles of any polygon add up to 360°, so the number of sides = 36024 = 15.

Question 21

A regular polygon has 54 diagonals. What is the measure of each of its interior angles?

A

144°✓✗

B

135°✓✗

C

156°✓✗

D

150°✓✗

Explanation

Number of diagonals = n(n - 3)2 = 54, so n² - 3n - 108 = 0, (n - 12)(n + 9) = 0 and n = 12.
Each exterior angle = 36012 = 30°, so each interior angle = 180 - 30 = 150°.

Question 22

An arc of a circle subtends an angle of 130° at the centre. What angle does it subtend at any point on the remaining part of the circle?

A

130°✓✗

B

65°✓✗

C

115°✓✗

D

50°✓✗

Explanation

The angle an arc subtends at the centre is twice the angle it subtends at any point on the remaining part of the circle.
Angle = 1302 = 65°.

Question 23

AB is a diameter of a circle and C is a point on the circle. If ∠CAB = 34°, what is ∠CBA?

A

34°✓✗

B

66°✓✗

C

56°✓✗

D

146°✓✗

Explanation

The angle in a semicircle is a right angle, so ∠ACB = 90°.
∠CBA = 180 - 90 - 34 = 56°.

Question 24

A point P is 25 cm from the centre of a circle of radius 7 cm. What is the length of the tangent drawn from P to the circle?

A

24 cm✓✗

B

18 cm✓✗

C

26 cm✓✗

D

32 cm✓✗

Explanation

The radius to the point of contact is perpendicular to the tangent, so the radius, the tangent and OP form a right triangle with OP as hypotenuse.
Tangent = √(25² - 7²) = √(625 - 49) = √576 = 24 cm.

Question 25

Two circles of radii 10 cm and 17 cm intersect each other, and the distance between their centres is 21 cm. What is the length of their common chord?

A

8 cm✓✗

B

12 cm✓✗

C

20 cm✓✗

D

16 cm✓✗

Explanation

The line joining the centres is the perpendicular bisector of the common chord. Let it meet the chord at a point x cm from the centre of the smaller circle, and let half the chord be h.
h² = 10² - x² = 17² - (21 - x)²
100 - x² = 289 - 441 + 42x - x², so 42x = 252 and x = 6.
h = √(100 - 36) = 8, so the common chord = 2 × 8 = 16 cm.