Question

If $ (3,\,\,3) $ is a vertex of a triangle and $ (-3,\,\,6) $ and $ (9,\,\,6) $ are the mid points of the two sides through this vertex, then the centroid of the triangle is

• A. $ (3,\,\,7) $
• B. $ (1,\,\,7) $
• C. $ (-3,\,\,7) $
• D. $ (-1,\,\,7) $

Answer 1

The Correct Option is A

Given, $ A=(3,3),E=(-3,6) $ and $ F=(9,6) $ Let $ B=({{x}_{1}},{{y}_{1}}) $ and $ C=({{x}_{2}},{{y}_{2}}) $
Then, $ \frac{{{x}_{1}}+3}{2}=-3,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{y}_{1}}+3}{2}=6 $
$ \Rightarrow $ $ {{x}_{1}}=-9,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{y}_{1}}=9 $
and $ \frac{{{x}_{2}}+3}{2}=9,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{y}_{2}}+3}{2}=6 $
$ \Rightarrow $ $ {{x}_{2}}=15,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{y}_{2}}=9 $
Now, centroid $ =\left( \frac{-9+15+3}{3},\frac{9+9+3}{3} \right) $
$ =(3,7) $

Answer 2

A triangle's centroid is the location inside the triangle where all of its medians meet or cross. Straight lines that connect a vertex to its opposite side and divide that side into two equal half are called medians. The triangle is divided into smaller, equal-sized triangles using a median as well. There are four locations when a triangle is concurrent.

Orthocentre

Incentre

Circumcentre 

Centroid

A triangle's centroid is always found inside the triangle's perimeter.

The centroid of the triangle is shown by point G in this instance. The centroid of a triangle always resides inside the triangle's perimeter, just like the incentre does. To get the centroid's coordinates, the section formula of coordinate geometry is used. A triangle's centroid, which coincides with its medians, always rests on a median and splits it 2:1. The centroid theorem refers to this.

The relative midpoints of the triangle's sides are denoted by D, E, and F. The triangle ABC's centroid is labeled G.

Following is a discussion of the characteristics of a triangle's centroid:

• The point where all three of a triangle's medians meet or converge is known as the centroid.
• The medians are split in half at the centroid in a 2:1 ratio. The centroid theorem is another name for this.
• The most frequent point of concurrency inside a triangle is its centroid. Its significance in both mathematics and physics stems from the fact that it is the triangle's geometric core.
• The centroid's coordinates satisfy the straight-line equations for the three medians since it is the intersection of all three of them. 
• It is always situated inside a triangle's rim.

The coordinates of the three vertices of the triangle ABC are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃). D, E, and F are the respective midpoints of the sides BC, AC, and AB. G is the centroid.

Then the coordinates of the centroid of the triangle ABC are given as :