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# Centroid, Incentre and Circumcentre and Orthocentre of a Triangle - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

## Quick Facts

• Centroid is considered one of the most asked concept.

• 16 Questions around this concept.

## Solve by difficulty

If the vertices $\mathrm{P, Q, R}$ of a triangle $\mathrm{P QR}$ are rational points, which of the following point(s) of the triangle is (are) always rational point(s)?

The coordinates of the middle points of the sides of a triangle are $\mathrm{(4,2)(3,3)}$ and $\mathrm{(2,2)}$  then the coordinates of its centroid is:

## Concepts Covered - 1

Centroid

Centroid

A median is the line joining the mid-points of a side and the opposite vertex of a triangle.

Centroid  of a triangle is the point of intersection of the three medians of the triangle. A centroid divides any median in the ratio 2:1.

The coordinates of the centroid of a triangle (G) whose vertices are A (x1, y1), B (x2, y2) and C(x3, y3), is given by

$\\\mathrm{\mathbf{\left ( \frac{x_1+x_2+x_3}{3},\;\frac{y_1+y_2+y_3}{3} \right )}}$

Note:

If D (a1, b1), E (a2, b2) and F (a3, b3) are the mid point of ΔABC, then centroid of triangle ABC is given by

$\\\mathrm{\mathbf{\left ( \frac{a_1+a_2+a_3}{3},\;\frac{b_1+b_2+b_3}{3} \right )}}$

Example

If Origin is the centroid of a triangle ABC, and the coordinates of other two vertices of the triangle are  A (4, –3) and B (–5, 2), then find the coordinates of the third vertex.

Solution

Let point C is  (α, β)

$\\\mathrm{(0,0)=\left (\frac{\alpha+4-5}{3} ,\frac{\beta-3+2}{3} \right )}\\\\\mathrm{\Rightarrow \frac{\alpha+4-5}{3}=0\;\;and\;\;\frac{\beta-3+2}{3}=0}\\\\\mathrm{\alpha=1,\;\;and\;\;\beta=1}$

Coordinates of C are (1, 1)

## Study it with Videos

Centroid

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## Books

### Reference Books

#### Centroid

Mathematics for Joint Entrance Examination JEE (Advanced) : Coordinate Geometry

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