Question

If orthocenter and circumcenter of a triangle are respectively (1, 1) and (3, 2), then the coordinates of its centroid are:
(a) \[\left( \dfrac{7}{3},\dfrac{5}{3} \right)\]
(b) \[\left( \dfrac{5}{3},\dfrac{7}{3} \right)\]
(c) (7, 5)
(d) None of these

Answer 1

Hint: While solving this question, we should remember that the centroid divides the orthocenter and the circumcenter in the ratio 2:1. We know the coordinates of the endpoints and also the ratio. Therefore, we can use the internal section formula to calculate the coordinates of the centroid.

Complete step-by-step answer:
The orthocenter and circumcenter of a triangle are (1, 1) and (3, 2) respectively. Let the orthocenter be O and the circumcenter be C.
O = (1, 1)
C = (3, 2)
The centroid divides the line between the orthocenter and the circumcenter in the ratio 2:1. Therefore, we can use the internal section formula to determine the coordinates of the centroid. We know that the internal section formula is given by,
\[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]
Here, let us consider that the centroid divides the orthocenter and the circumcenter in the ratio m:n.
m:n = 2:1
Therefore, m = 2 and n = 1. Let us assume the orthocenter having coordinates \[\left( {{x}_{1}},{{y}_{1}} \right)\] and circumcenter having coordinates \[\left( {{x}_{2}},{{y}_{2}} \right).\]
\[O=\left( 1,1 \right)=\left( {{x}_{1}},{{y}_{1}} \right)\]
\[C=\left( 3,2 \right)=\left( {{x}_{2}},{{y}_{2}} \right)\]
Let us assume the centroid as G. On substitution, we get,
\[G\left( x,y \right)=\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]
Substituting the values of m, n, \[{{x}_{1}}\text{ and }{{x}_{2}}\] in the above expression, we get,
\[G\left( x,y \right)=\left( \dfrac{2\times 3+1\times 1}{2+1},\dfrac{2\times 2+1\times 1}{2+1} \right)\]
Solving the equation further, we get,
\[G\left( x,y \right)=\left( \dfrac{6+1}{3},\dfrac{4+1}{3} \right)\]
\[G\left( x,y \right)=\left( \dfrac{7}{3},\dfrac{5}{3} \right)\]
Therefore, \[\left( \dfrac{7}{3},\dfrac{5}{3} \right)\] are the coordinates of the centroid.
Hence, option (a) is the right answer.

Note: Students usually get confused between the orthocenter, circumcenter, and the centroid of a triangle. Remember that the orthocenter is the point of intersection of the three heights of a triangle. A circumcenter is the point of intersection of the three perpendicular bisectors of a triangle. A centroid is a point of intersection of the three medians of a triangle.