All Exams  >   SSC CGL  >   Quantitative Aptitude for SSC CGL  >   All Questions

All questions of Geometry for SSC CGL Exam

Four horses are tethered at four comers of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.
  • a)
    42 m2
  • b)
    22 m2
  • c)
    84 m2
  • d)
    168 m2
Correct answer is option 'B'. Can you explain this answer?

Preeti Khanna answered
Total area of plot = 14 * 14 = 196m2
Horses can graze in quarter circle of radius = 7m
Grazed area = 4 * (pie r2)/4 = 154 m2
Area of plot when horses cannot reach = (196 - 154) = 42m2
Ungrazed area = 42 - 20 = 22m2

A cyclic quadrilateral is such that two of its adjacent angles are divisible by 6 and 10 respectively. One of the remaining angles will necessarily be divisible by:
  • a)
    3
  • b)
    4
  • c)
    8
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Palak Bose answered
We know that the sum of the opposite angles of a cyclic quadrilateral is 180 degrees. Let the four angles be A, B, C, and D, with A and B being the angles divisible by 6 and 10, respectively.

Since A is divisible by 6 and B is divisible by 10, we know that A = 6m and B = 10n for some integers m and n.

Now, consider the opposite angles. Since the sum of opposite angles is 180 degrees, we have:

C = 180 - B = 180 - 10n
D = 180 - A = 180 - 6m

We want to find which of the given options the angles C or D are necessarily divisible by. Let's examine each option:

1. 3: Since B is divisible by 10, it is possible that B is divisible by 5 but not 3 (e.g. B = 10). In this case, C = 180 - B would not be divisible by 3. Also, A is divisible by 6, so A is always divisible by 3, which means D = 180 - A would never be divisible by 3. So, this option is incorrect.

2. 4: Since A is divisible by 6, it is possible that A is divisible by 2 but not 4 (e.g. A = 6). In this case, D = 180 - A would not be divisible by 4. Also, B is divisible by 10, so B is always divisible by 2, which means C = 180 - B would never be divisible by 4. So, this option is also incorrect.

3. 8: If A is divisible by 6, then it can be even or odd multiples of 6 (e.g. A = 6, 12, 18, ...). D will be 180 - A, which means D can be both even and odd (e.g. D = 180 - 6 = 174, D = 180 - 12 = 168, D = 180 - 18 = 162, ...). Since D can be both even and odd, it is not necessarily divisible by 8. Similarly, C can also be both even and odd, so it is not necessarily divisible by 8. Thus, this option is also incorrect.

4. None of these: Since none of the previous options work, the correct answer is None of these.

So, the correct answer is option 4: None of these.

The ratio of the sides of Δ ABC is 1:2:4. What is the ratio of the altitudes drawn onto these sides?
  • a)
    4:2:1
  • b)
    1:2:4
  • c)
    1:4:16
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Aakash Giery answered
Sum of any two sides should be greater than third side.
here 1+2=3 is not less than 4 ,
1+2<4 ,so="" triangle="" is="" not="" possible.="" ,so="" triangle="" is="" not="">

A, B, C are the three angles of a Δ. If A − B = 15° and B − C = 30°. Then ∠A is equal to :
  • a)
    65°
  • b)
    80°
  • c)
    75°
  • d)
    85°
Correct answer is option 'B'. Can you explain this answer?

Ssc Cgl answered
Since A, B and C are the angles of a Δ,
∴ A + B + C = 180° .................... (1)
According to question,
A – B = 15° ;
⇒ A = B + 15°...................(2)
B – C = 30°;
⇒ B = C + 30°;....................(3)
Put the value of B from equation (2) in Equation (1), we will get
∴ A = B + 15°
A = C + 30° + 15°
A = C + 45° ......................(4)
From a equation,
∴ A + B + C = 180°
⇒ (C + 45°) + (C + 30°) + C = 180°
⇒ 3C + 45° + 30° = 180°
⇒ 3C = 180° – 75° = 105°
⇒ C = 35° ...........................(5)
From equation (4)
A = C + 45°
Put the value of C from equation (5) , we will get
∴ ∠A = 35° + 45° = 80°.

Find the value of x in the given figure.
  • a)
    16 cm  
  • b)
    7 cm
  • c)
    12 cm  
  • d)
    9 cm
Correct answer is option 'D'. Can you explain this answer?

Pooja Sen answered
Isosceles trapezium is always cyclic The sum of opposite angles of a cyclic quadrilateral is 180°

The area of similar triangles, ABC and DEF are 144cm2 and 81 cm2 respectively. If the longest side of the larger △ABC be 36 cm, then the longest side of the smaller △DEF is:
  • a)
    27 cm 
  • b)
    26 cm
  • c)
    29 cm 
  • d)
    30 cm
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given:
- The area of triangle ABC is 144 cm^2.
- The area of triangle DEF is 81 cm^2.
- The longest side of triangle ABC is 36 cm.

To find:
- The length of the longest side of triangle DEF.

Explanation:

Step 1: Finding the Scale Factor
- The area of a triangle is given by the formula: Area = (1/2) * base * height.
- Since the triangles ABC and DEF are similar, their areas are proportional to the square of their corresponding sides.
- Therefore, we can write: (AB/DE)^2 = Area of ABC/Area of DEF.

- Substituting the given values, we get: (AB/DE)^2 = 144/81.

- Simplifying the equation, we have: (AB/DE)^2 = 16/9.

- Taking the square root of both sides, we get: AB/DE = √(16/9).

- Simplifying further, we have: AB/DE = 4/3.

- This ratio represents the scale factor between the two triangles.

Step 2: Finding the Length of the Longest Side of Triangle DEF
- The longest side of triangle ABC is given as 36 cm.

- Using the scale factor, we can write: AB/DE = 4/3.

- Substituting the values, we have: 36/DE = 4/3.

- Cross-multiplying, we get: 4DE = 36 * 3.

- Simplifying the equation, we have: 4DE = 108.

- Dividing both sides by 4, we get: DE = 27.

- Therefore, the length of the longest side of triangle DEF is 27 cm.

Conclusion:
- The length of the longest side of triangle DEF is 27 cm.
- Hence, the correct answer is option A.

An angle is equal to one-third of its supplement. Its measure is equal to :
  • a)
    40°
  • b)
    50°
  • c)
    45°
  • d)
    55°
Correct answer is option 'C'. Can you explain this answer?

Ssc Cgl answered
Let the measured of the required angle be P degree.
Then, its supplement = 180 – P
Now use the formula,

3P + P180°
⇒ P = 45°

The volume of two spheres are in the ratio 27 : 125. The ratio of their surface area is?
  • a)
    25 : 9
  • b)
    27 : 11
  • c)
    11 : 27
  • d)
    9 : 25
Correct answer is option 'D'. Can you explain this answer?

Sagar Sharma answered
Understanding the Volume and Surface Area of Spheres
The problem states that the volumes of two spheres are in the ratio 27:125. To find the ratio of their surface areas, we need to understand the relationships between the two.
Volume of a Sphere
- The formula for the volume (V) of a sphere is given by V = (4/3)πr^3, where r is the radius.
- If the volumes of two spheres are in the ratio 27:125, we can express this as:
- V1/V2 = 27/125
Finding the Ratio of Radii
- Since volumes are proportional to the cube of the radii, we have:
- (r1^3)/(r2^3) = 27/125
- Taking the cube root on both sides gives us:
- r1/r2 = (27^(1/3))/(125^(1/3)) = 3/5
Surface Area of a Sphere
- The formula for the surface area (A) of a sphere is A = 4πr^2.
- Now, to find the ratio of the surface areas of the two spheres, we have:
- A1/A2 = (4πr1^2)/(4πr2^2) = (r1^2)/(r2^2)
Calculating the Surface Area Ratio
- Substituting the ratio of the radii:
- r1/r2 = 3/5
- Therefore, (r1^2)/(r2^2) = (3^2)/(5^2) = 9/25
Final Answer
- The ratio of the surface areas of the two spheres is 9:25, which corresponds to option 'D'.

The complement of 30°20′ is:
  • a)
    69°40′
  • b)
    59°40′
  • c)
    35°80′
  • d)
    159°40′
Correct answer is option 'B'. Can you explain this answer?

EduRev SSC CGL answered
Complement of 30°20′ = 90° – ( 30°20′ ) = 90° – ( 30° + 20′ )
= (89° – 30°) + (1° – 20′)
= 59° + 60′ – 20′ [ ∴ 1° = 60°′]
= 59° + 40′ = 59°40′.

A pond 100 m in diameter is surrounded by a circular grass walk-way 2 m wide. How many square metres of grass is the on the walk-way?
  • a)
    98 π
  • b)
    100 π
  • c)
    204 π
  • d)
    202 π
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
The radius of the pond is 100/2 = <100 =50="">>50 m.
The radius of the grass walkway is 50+2 = <50+2=52>>52 m.
The area of the grass walkway is pi*(52^2 - 50^2) = pi*(2704 - 2500) = pi*204
≈ 204*pi
≈ 204*3.14
≈ <204*3.14=640.56>>640.56 m^2.
So, the answer is a) 640.

In the given figure, AD is the bisector of ∠BAC, AB = 6 cm, AC = 5 cm and BD = 3 cm. Find DC. It is given that ∠ABD = ∠ACD.
  • a)
    11.3 cm 
  • b)
    4 cm
  • c)
    3.5 cm 
  • d)
    2.5 cm
Correct answer is option 'D'. Can you explain this answer?

Pooja Shah answered
We know that the internal bisector of angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
Hence:
In triangle ABD and ACD
Angle BAD = CAD (Given AD is the bisector)
Angle ABD = ACD (GIven)
there fore they are similar (AAA Property)
AB/BD = AC/CD
6/3 = 5/CD
CD = 2.5 cm

In a triangle ABC, incentre is O and ∠BOC = 110o, then the measure of ∠BAC
  • a)
    20o
  • b)
    40o
  • c)
    55o
  • d)
    110o
Correct answer is option 'B'. Can you explain this answer?

Ssc Cgl answered
As given,
∠BOC = 110o
And, we know incentre of a triangle is a point where angle bisector of triangles meet.
So, which means ∠OCB =∠OBC
∠BOC +∠OCB+∠OBC = 180o
⇒∠OCB + ∠OBC = 180o − 110o
⇒∠OCB +∠OBC = 70o

Since, ∠ABC = 2∠OBC
∠ACB = 2∠OCB
So, ∠ABC+∠ACB = 2 × 70o
⇒∠ABC + ∠ACB = 140o
Now, in triangle ABC.
∠ABC +∠BAC +∠ACB = 180o
⇒∠BAC  +140o = 180o
⇒∠BAC = 180o − 140o
⇒∠BAC = 40o
Hence, the correct answer is 40o.

The line AB is 6 m, in length and is tangent to the inner of the two concentric circles at point C. It is known that the radii of the two circles are integers. The radiusof the outer circle is-------, where A and B are points on the outer circle.
  • a)
    5 m
  • b)
    4 m
  • c)
    6 m
  • d)
    3 m
Correct answer is option 'A'. Can you explain this answer?

Aarav Sharma answered
Given Information

- The line AB is 6 m in length and is tangent to the inner of the two concentric circles at point C.
- The radii of the two circles are integers.

To Find

- The radius of the outer circle.

Solution

Let's consider the diagram below:

We can see that the radius of the inner circle is equal to the length of the perpendicular from the center of the circles to the line AB.

Let O be the center of the circles, and let OC be x. Then, by the Pythagorean theorem, we have:

OA^2 = OC^2 + AC^2

OB^2 = OC^2 + BC^2

Since OA = OB (both A and B are on the outer circle), we can subtract the two equations to get:

AC^2 - BC^2 = 0

(AC + BC)(AC - BC) = 0

Since AC and BC are both positive, we have AC = BC.

Therefore, the line AB is equidistant from A and B, which means it passes through the center O of the circles.

Let r be the radius of the outer circle. Then, we have:

OC = r - x (since x is the radius of the inner circle)

By the Pythagorean theorem, we also have:

AC^2 = r^2 - OC^2 = r^2 - (r - x)^2 = 2rx - x^2

Since AC = BC, we have:

2rx - x^2 = (6/2)^2 = 9

Simplifying, we get:

x(2r - x) = 9

Since x is an integer, and 2r - x is also an integer, we can see that x must be a factor of 9. The possible values of x are:

x = 1, 3, or 9

If x = 1, then 2r - x = 2r - 1 is odd, which means r is not an integer.

If x = 3, then 2r - x = 2r - 3 is odd, which means r is not an integer.

Therefore, we must have x = 9, which gives:

2r - x = 2r - 9

x(2r - x) = 9

Substituting x = 9, we get:

r = (x^2 + 9)/2x = (81 + 9)/18 = 5

Therefore, the radius of the outer circle is 5 m.

Answer: (a) 5 m

In a ΔABC, if 2∠A = 3∠B = 6∠C, Then ∠A is equal to:
  • a)
    60°
  • b)
    30°
  • c)
    90°
  • d)
    120°
Correct answer is option 'C'. Can you explain this answer?

Malavika Rane answered
Given Information:
In triangle ΔABC, 2∠A = 3∠B = 6∠C

Solution:

Step 1: Finding the sum of angles in a triangle
In a triangle, the sum of all interior angles is always 180°.

Step 2: Expressing angles in terms of ∠A
Given, 2∠A = 3∠B = 6∠C
Let's express angles B and C in terms of angle A:
2∠A = 3∠B
∠B = 2/3∠A
Also,
2∠A = 6∠C
∠C = 1/3∠A

Step 3: Substituting angle values in the sum of angles formula
Now, substitute the expressions for ∠B and ∠C in terms of ∠A into the formula for the sum of angles in a triangle:
∠A + ∠B + ∠C = 180°
∠A + 2/3∠A + 1/3∠A = 180°
(6/3)∠A = 180°
2∠A = 180°
∠A = 90°
Therefore, ∠A is equal to 90° (option c).

AB is the diameter of the circle and ∠PAB=40∘
what is the value of ∠PCA?
  • a)
    50∘
  • b)
    55°
  • c)
    70° 
  • d)
    45°
Correct answer is option 'A'. Can you explain this answer?

  • In △PAB
    ⇒  ∠PAB=40o         [ Given ]
    ⇒  ∠BPA=90o      [ angle inscribed in a semi-circle ]
    ⇒  ∠PAB+∠PBA+∠BPA=180o
    ∴   40o+∠PBA+90o=180o
    ∴   ∠PBA=180o−130o
    ∴   ∠PBA=50o
    ⇒  ∠PBA=∠PCA=50o     [ angles inscribed in a same arc PA ] 
    ∴   ∠PCA=50o

Chapter doubts & questions for Geometry - Quantitative Aptitude for SSC CGL 2026 is part of SSC CGL exam preparation. The chapters have been prepared according to the SSC CGL exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for SSC CGL 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.

Chapter doubts & questions of Geometry - Quantitative Aptitude for SSC CGL in English & Hindi are available as part of SSC CGL exam. Download more important topics, notes, lectures and mock test series for SSC CGL Exam by signing up for free.

Top Courses SSC CGL