Question

The centroid of a triangle is (2,7) and two of its vertices are (4,8) and (-2,6). The third vertex is:
(a) (0,0)
(b) (4,7)
(c) (7,4)
(d) (7,7)
(e) (4,4)

Answer 1

Hint: In this question, let us assume the coordinates of the other vertex and then find the midpoint of the given two vertices. Now, the centroid is the point which divides the third vertex and this mid-point in the ratio 2:1. Then by applying the section formula we can get the third vertex.

\[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]

CENTROID: Median is the line joining the midpoint of a side to the opposite vertex. The three medians of a triangle meet at a point is called the centroid. Centroid divides the median in the ratio 2:1.

Complete step-by-step answer:


Let us assume the other vertex as (x, y).


Now, let us find the midpoint of the given two vertices.


As we already know that the midpoint of the points joining (m, n) and (l, k) is given by:
\[\left( \dfrac{m+l}{2},\dfrac{n+k}{2} \right)\]


Now, by using the same formula we can get the midpoint of the given two vertices.


Given , vertices are (4,8) and (-2,6)


\[\Rightarrow \left( \dfrac{4+\left( -2 \right)}{2},\dfrac{8+6}{2} \right)\]


Now, this can be further simplified as.


\[\begin{align}


  & \Rightarrow \left( \dfrac{2}{2},\dfrac{14}{2} \right) \\


 & \Rightarrow \left( 1,7 \right) \\


\end{align}\]


Now, the centroid divides the points (x, y) and (1,7) in the ratio 2:1.


From, the section formula we have,


The coordinate of the point which divides the joint of (x1, y1) and (x2, y2) in the ratio m:n internally is:


\[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]


Given, the centroid is (2,7) and the points are (x, y) and (1,7) and the ratio is 2:1.


Now, on comparing these values with the above formula we get,


\[\begin{align}


  & m=2,n=1 \\


 & {{x}_{1}}=x,{{x}_{2}}=1 \\


 & {{y}_{1}}=y,{{y}_{2}}=7 \\


\end{align}\]


Let us now substitute these values in the respective formula.


\[\Rightarrow \left( 2,7 \right)=\left( \dfrac{2\times 1+1\times x}{2+1},\dfrac{2\times


7+1\times y}{2+1} \right)\]


Now, on further simplification we get,


\[\Rightarrow \left( 2,7 \right)=\left( \dfrac{2+x}{3},\dfrac{14+y}{3} \right)\]


Let us now multiply with 3 on both sides.


\[\Rightarrow \left( 6,21 \right)=\left( 2+x,14+y \right)\]


Now, on equating the respective coordinates on the both sides we get,


\[\begin{align}


  & \Rightarrow 6=2+x \\


 & \therefore x=4 \\


\end{align}\]


\[\begin{align}


  & \Rightarrow 21=14+y \\


 & \therefore y=7 \\


\end{align}\]


Thus, the third vertex of the given triangle is (4,7)


Hence, the correct option is (b).

Note: Instead of finding the midpoint of the given two vertices and then by using the section formula to get the other vertex we can directly use the formula for the centroid of a triangle with given three vertices. Then further simplify it to get the coordinates of the third vertex. Both the methods give the same result.
\[G=\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]
While applying the section formula we use the formula that divides the given two points in the given ratio internally because here centroid lies between the midpoint of two vertices and the third vertex. Here, if we use the externally dividing formula then the result will change completely.
It is important to note that while substituting the coordinates we need to substitute them accordingly without interchanging the terms and without neglecting the signs because it changes the answer.