Centroid MCQ Quiz - Objective Question with Answer for Centroid - Download Free PDF

Concept:

• Median of a triangle: The line segment joining a vertex to the midpoint of the opposite side.
• Centroid (G): The point of intersection of medians; it divides each median in the ratio 2:1.
• Circle touches triangle side: Use geometrical relationships based on circle properties and distances.
• Important property: If AG:AF = AB², and AG = 2×GD, then algebraic equations help to find required lengths.

Calculation:

Let GD = x, DF = y

⇒ AG = 2x, AF = x + y

⇒ 2x(3x + y) = 36

⇒ xy = 4

⇒ 3x² + 4 = 18

⇒ x² = 14/3

⇒ AD = 3x = √42

Now, AC² + AB² = 2(AD² + BD²)

⇒ AC² + 36 = 2(42 + 4)

⇒ AC² + 36 = 92

⇒ AC² = 56

AC² = 56

Concept:

• Median of a triangle: The line segment joining a vertex to the midpoint of the opposite side.
• Centroid (G): The point of intersection of medians; it divides each median in the ratio 2:1.
• Circle touches triangle side: Use geometrical relationships based on circle properties and distances.
• Important property: If AG:AF = AB², and AG = 2×GD, then algebraic equations help to find required lengths.

Calculation:

Let GD = x, DF = y

⇒ AG = 2x, AF = x + y

⇒ 2x(3x + y) = 36

⇒ xy = 4

⇒ 3x² + 4 = 18

⇒ x² = 14/3

⇒ AD = 3x = √42

AD = √42

AD² = 42

Calculation 

Let ‘G’ be the centroid of Δ formed by (1, 3) (3, 1) & (2, 4)

Image of G w.r.t. x + 2y – 2 = 0

= 

⇒ 

15(α – β) = – 2 + 24 = 22

Hence option 4 is correct

Concept:

To reflect a point    across the line ax + by + c = 0, the reflected point (x', y') is given by:

Explanation:  

Vertex 1: (1, 3)

  

Reflected point: (-1, -1) 

Vertex 2: (3, 1)

   

Reflected point:   
Vertex 3: (2, 4)

Reflected point:  

Centroid of   :

Substitute the reflected points:

 


  

   


   

   

Hence 22 is the correct answer.

Concept Used:

The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by

The distance between two points (x₁, y₁) and (x₂, y₂) is given by

Calculation:

Let the coordinates of point C be (x, y).

Centroid of triangle ABC = =

Distance of the centroid from the origin = 5 units

⇒ = 5

⇒ = 25

⇒ = 225

⇒ = 225

⇒ = 0

This equation represents a circle.

To find the radius, we compare it with the general equation of a circle:

Here, 2g = 10 ⇒ g = 5

2f = 10 ⇒ f = 5

c = -175

Radius = = = = 15

Diameter = 2 × radius = 2 × 15 = 30 units

∴ The locus of C is a circle with a diameter of 30 units.

Hence option 3 is correct.

Concept:

Centroid: The point at which the three medians of the triangle intersect is known as the centroid of a triangle.

If the three vertices of the triangle are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃), then the

Calculation:

Given:

A, B, and C are vertices of the triangle

Position vectors of A, B, and C be  .

G is the centroid of △ABC

∴ G = 

⇒  =  

⇒  = 

⇒ = 

∴​  = 

Concept:

Centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x1, y1), B(x2, y2), C(x3, y3).

Centroid of a triangle = 

Calculation:

Given vertices are (2, 3), (-2, -5) and (3, 5)

Centroid of a triangle =  = (1, 1)

Concept:

The centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x1, y1), B(x2, y2), C(x3, y3)

The centroid of a triangle

=

Calculation:

Here, vertices of triangle (4,x), (y,-5) and (7, 8) and Centroid = (3, 5)

So, 3 = 

⇒11+y = 9 

⇒ y = -2

And, 5 = 

⇒ x + 3 = 15

⇒ x = 12

∴ x = 12, and y = -2

Hence, option (3) is correct. 

Concept:

The centroid of the triangle formed by joining three points (x₁, y₁), (x2, y2) and (x3, y3) is given by .

Calculation:

The centroid of the triangle formed by joining the three points (3, 0), (6, 2) and (2, 3), will be:

= .

Concept:

Centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x₁, y₁), B(x₂, y₂), C(x₃, y₃).

Centroid of a triangle = 

Calculation:

Vertices of the triangle are given as A(2, 4), B(1, 0) and C(0, -1)

Centroid of a triangle = 

Concept:

If A (x₁, y₁, z₁), B (x₂, y₂, z₂) and C (x₃, y₃, z₃) are the vertices of ΔABC. Then the centroid of ΔABC is given by:

Calculation:

Given: The vertices of ΔABC are: A(2, -3, 3), B(5, -3, -4) and C(2, -3, -2)

As we know that, if A (x₁, y₁, z₁), B (x₂, y₂, z₂) and C (x₃, y₃, z₃) are the vertices of ΔABC. Then the centroid of ΔABC is given by:

CONCEPT:

If the coordinates of the vertices of triangle Δ ABC are: A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃). Then the co-ordinates of the centroid G is given by:

CALCULATION:

Given: (2, 7) is the centroid of the triangle whose two vertices are (4, 8) and (-2, 6).

Here, we have to find the third vertex of the given triangle.

Let A = (4, 8), B = (2, 6) and C = (x, y) be the three vertices of triangle ABC whose centroid is G = (2, 7)

As we know that, if  A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC then the co-ordinates of the centroid G is given by: 

Here, x₁ = 4, y₁ = 8, x₂ = - 2, y₂ = 6, x₃ = x and y₃ = y

So, the centroid of triangle ABC is : 

∵  G = (2, 7)

⇒ 

⇒ x = 4 and y = 7

So, the third vertex is (4, 7)

Hence, option B is true.

Given: 

Vertices of triangle are (4, p), (-1, -1) and (3, 5)

Concept:

Centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x1, y1), B(x2, y2), C(x3, y3).

The centroid of a triangle = 

Mid-point: If two points are (x₁, y₁) and (x₂, y₂), then mid point is given by

Calculation:

We know that, centroid of triangle is given by 

G = 

Therefore, for given triangle, we will get centroid 

G = 

According to question, G is the mid-point of (1, 4) & (q, 2). Therefore, by using mid-point formula

On comparing both side

⇒ q = 3 & q = 5

⇒ p² + q² = 34

Hence, option 4 is correct.

Concept:

Centroid of a Triangle: The point where the three medians of the triangle intersect.

Let’s consider a triangle. If the three vertices of the triangle are A(x₁, y₁), B(x₂, y₂), C(x₃, y₃).

Centroid of a triangle = 

Calculation:

Let’s consider a triangle. If the three vertices of the triangle are A(x₁, y₁), B(x₂, y₂), C(x₃, y₃).

Given: A (x₁, y₁) = A (1, 1)

Mid-Point of AB is given (-1, 2)

⇒ x₂ = -2 – 1 = -3 and y₂ = 4 – 1 = 3

So, B(x₂, y₂) = B (-3, 3)

Mid-Point of AC is given (3, 2)

⇒ x₃ = 6 – 1 = 5 and y₃ = 4 – 1 = 3

C(x₃, y₃) = (5, 3)

Centroid of a triangle = 

Concept:

Centroid of a triangle:

If the three vertices of the triangle are A(x1, y1), B(x2, y2), C(x3, y3) then the centroid of a triangle = 

Calculation:

Given: P(1, 3), Q(3, 6) and R(- 4, 0)

As we know that, the centroid of a triangle whose vertices are A(x1, y1), B(x2, y2), C(x3, y3) is given by: 

Let the centroid of a triangle be S.

Hence, the correct option is 2.