Centroid MCQ Quiz in मराठी - Objective Question with Answer for Centroid - मोफत PDF डाउनलोड करा

Calculation

Centroid G divides MR in 1 ∶ 2   

By using section formula 

⇒ G(1, 2, 2) 

⇒ x = 2

⇒ x = 1 + m ⇒ 2 = 1 + m ⇒ m = 1

⇒ y = -6, z = 0

Point of intersection A of given lines is (2,–6, 0) 

Hence option 3 is correct

Calculation

Given 

L: 

Point on line L = P(8k - 3, 2k + 4, 2k - 1)

Distance of P from R(1,2,3) is 6 units

⇒ 

⇒ 

⇒ 72k² - 72k + 36 = 36

⇒ 72k2 - 72k = 0

⇒ k = 0,1

At k = 0 ⇒ P(-3,4,-1)

At k = 1 ⇒ Q(5,6,1)

The centroid of the triangle PQR is 

⇒ 

⇒ (1,4,1) = (α, β, γ)

α2 + β2 + γ2 = 1 + 16 + 1 = 18

Hence option 3 is correct

Given :

(a cos θ1, a sin θ1), (a cos θ2, a sin θ2) and (a cos θ3, a sin θ3) represents the vertices of an equilateral triangle which is inscribed in a circle.

Formula used :

Centroid (x, y) =          ------ (1)

Calculations :

We know, equilateral triangle is inscribed in a circle with centre (0, 0)

We know from the above figure that

M is the circumcentre of the equilateral 

M is also the centroid of 

Using equation (1), we get

⇒ 

⇒ 

⇒ 

∴ Both statements I and II are correct.

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: (1, 2), (h, - 3) and (- 4, k) are the vertices of triangle and  the centroid of the triangle is located at (5, - 1).

Let A = (1, 2), B = (h, - 3) and C = (- 4, k) are the vertices of Δ ABC.

As we know that, the centroid of a triangle is given by: 

Here, x₁ = 1, y₁ = 2, x₂ = h, y₂ = - 3, x₃ = - 4 and y₃ = k

⇒ 

So, the cenroid of the given triangle is: 

∵ It is given that the centroid of the triangle is located at (5, - 1).

⇒ 

⇒ h = 18 and k = - 2

⇒ h + k = 16

Hence, option A is the correct answer.

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 (7, x), (y, -6) and (9, 10)

Centroid of a triangle = (6, 3)

⇒ Centroid of a triangle = (6, 3) = 

⇒ (18, 9) = [(16 + y), (x + 4)]

∴ 16 + y = 18 and x + 4 = 9

⇒ y = 2 and x = 5

So, (x, y) = (5, 2)

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.

Calculation:

Let the coordinates of C be C = (x, y, z)

Given, A(3, 1, 4) and B(-4, 5, −3) and the centroid of the triangle is G(-1, 2, 1)

∴  = - 1

⇒ x = - 2

Also,  = 2

⇒ y = 0

and,  = 1

⇒ z = 2

∴ C = (x, y, z) = (- 2, 0, 2)

∴ The coordinates of C are (-2, 0, 2)

The correct answer is Option 2.

Calculation

ΔABC is equilateral

Orthocentre and centroid will be same

⇒ 

Mid - point of AB is D 

∴ ℓ₁ = 

⇒ ℓ1 = 

∴  

Hence option (2} is correct

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: A(4,4), B(-1,-2) and C(6,-8) are the coordinates of the triangle ABC.

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

Here, x₁ = 4, y₁ = 4, x₂ = - 1, y₂ = 2, x₃ = 6 and y₃ = 8.

So, the centroid of triangle ABC is: 

=  

So, the centroid of the given triangle ABC is (3, - 2)

Important Points

The Centroid of a triangle separates the medians in the ratio of 2 : 1

The Centroid of a triangle: The centroid of a triangle is formed when three medians of a triangle intersect. It is one of the four points of concurrency of a triangle. The medians of a triangle are constructed when the vertices of a triangle are joined with the midpoint of the opposite sides of the triangle.