Question

If the vertices of a triangle are \[(a,1),(b,3)\] and \[(4,c)\], then find the condition such that the centroid of the triangle will lie on the x-axis.
A.\[c = - 4\]
B.\[a + b + c = 0\]
C.\[a = - 2\]
D.\[b = - 4\]

Answer 1

Hint: Add the x coordinates of the given points to obtain the x coordinate of the centroid of the triangle, similarly add y to obtain the y of the centroid. Then equate the y coordinate of the obtained centroid to zero to obtain the required condition.

Formula Used:
The formula of the centroid of the triangle is,
\[\left( {\dfrac{{{a_1} + {a_2} + {a_3}}}{3},\dfrac{{{b_1} + {b_2} + {b_3}}}{3}} \right)\] , where three vertices of the triangle are \[\left( {{a_1},{b_1}} \right),\left( {{a_2},{b_2}} \right),\left( {{a_3},{b_3}} \right)\].

Complete step by step solution:
The centroid is \[\left( {\dfrac{{a + b + 4}}{3},\dfrac{{1 + 3 + c}}{3}} \right)\] .
Now, any point on the x-axis can be written as \[(x,0)\] as the displacement in the y-axis is 0 and the point only moves through the x-axis.
Therefore,
\[\dfrac{{c + 4}}{3} = 0\]
\[\Rightarrow c = - 4\]

Hence the correct option is A.

Additinal Information: The median is the line that joins the middle point of every vertex with the opposite vertex of the triangle. Three medians of the triangle divide the triangle into six equal parts. The point at which the three medians of the triangle intersect is the centroid of the triangle. Basically, the point of intersection of the medians of the triangle is the centroid. The centroid of a triangle always lies inside the triangle irrespective of the shape of the triangle. If three vertices of a triangle are given we can find the positions of the median and centroid easily.

Note: Sometime students get confused that the x coordinate or the y coordinate should be equal to zero to obtain the points on the x-axis, so the points on the x-axis are of the form \[(a,0)\] therefore, the y coordinate should be equal to zero.