Question

If the centroid of a triangle is at (1, 3) and two of its vertices are (-7, 6) and (8, 5) then find the third vertex of the triangle.

Answer 1

Hint: A centroid of a triangle is the point where the three line segments from one vertex to midpoint on the opposite side of the triangle meet.Given two vertices and centroid of the triangle. Use the formula of centroid to find the third vertex of the triangle i.e. Centroid=\[\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]

Complete step-by-step answer:
We have been given 2 vertices of a triangle i.e. (-7, 6) and (8, 5).
The centroid of the triangle is given as (1, 3).
Thus for a triangle, if the coordinates are given then the centroid of a triangle can be found by using a formula. If 3 vertices of a triangle are for example \[\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right)\] and \[\left( {{x}_{3}},{{y}_{3}} \right)\], then we can find the centroid by,
\[\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\] = centroid.
Let us assume that, \[\left( {{x}_{1}},{{y}_{1}} \right)=\left( -7,6 \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)=\left( 8,5 \right)\] and centroid is (1, 3). Thus substituting all these values in the equation of the centroid, we get
\[\left( \dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\dfrac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\] = centroid
\[\begin{align}
  & \left( \dfrac{-7+8+{{x}_{3}}}{3},\dfrac{6+5+{{y}_{3}}}{3} \right)=\left( 1,3 \right) \\
 & \left( \dfrac{1+{{x}_{3}}}{3},\dfrac{11+{{y}_{3}}}{3} \right)=\left( 1,3 \right) \\
\end{align}\]
Thus we can split it as, \[\dfrac{1+{{x}_{3}}}{3}=1\] and \[\dfrac{11+{{y}_{3}}}{3}=3\].
\[\therefore 1+{{x}_{3}}=3\]
Cross multiply and simplify it,
\[\begin{align}
  & {{x}_{3}}=3-1=2 \\
 & \therefore {{x}_{3}}=2 \\
\end{align}\]
Similarly, \[\dfrac{11+{{y}_{3}}}{3}=3\]
Cross multiply and simplify it,
\[\begin{align}
  & 11+{{y}_{3}}=9 \\
 & {{y}_{3}}=9-11=-2 \\
 & \therefore {{y}_{3}}=-2 \\
\end{align}\]
Thus we got the third vertex as \[\left( {{x}_{3}},{{y}_{3}} \right)=\left( 2,-2 \right)\].

Note: This question is the direct application of the values in the formula of the centroid. So it's important that you remember the basic formula of triangles to solve the problem easily.In some questions they give three vertices of the triangle and they ask to find the centroid of the triangle ,So by using formula we can calculate it.