Question

A triangle has corners at $\left( {1,3} \right),\,\left( {2, - 4} \right)\,and\,\left( {8, - 5} \right)$. If the triangle is reflected across the x-axis, what will its new centroid be?

Answer 1

Hint: In this question, it is given that the triangle is reflected across the x-axis. It means that the sign of the x-coordinate gets reversed but the sign of the y-coordinate remains unchanged. Then we have to use the formula $x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}$ and $y = \dfrac{{{y_1} + {y_2} + {y_3}}}{3}$to find the new centroid of the triangle.

Complete step by step answer:
In the above question, it is given that the coordinates of the triangle are $\left( {1,3} \right),\,\left( {2, - 4} \right)\,and\,\left( {8, - 5} \right)$. First, we have to find the reflected coordinates of the triangle. We know that after reflection from the x-axis the sign of the y-coordinate gets reversed and the sign of the x-coordinate remains unchanged.Therefore, the new coordinates of the triangle are $\left( {1, - 3} \right),\,\left( {2,4} \right)\,and\,\left( {8,5} \right)$.

Now, put the values ${x_1} = 1\,,\,{x_2} = \,2\,,\,{x_3} = 8\,\,and\,{y_1} = - 3\,,\,{y_2}\, = \,4\,,\,{y_3} = \,5$. On substituting the above values in formula $x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}$ and $y = \dfrac{{{y_1} + {y_2} + {y_3}}}{3}$, we get
$ \Rightarrow x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}$
$ \therefore x = \dfrac{{1 + 2 + 8}}{3}$
On simplification, we get
$ \Rightarrow x = \dfrac{{11}}{3}$
Now, we will find the y-coordinate
$ \Rightarrow y = \dfrac{{{y_1} + {y_2} + {y_3}}}{3}$
$ \Rightarrow y = \dfrac{{ - 3 + 4 + 5}}{3}$
$ \Rightarrow y = \dfrac{6}{3}$
$ \therefore y = 2$

Therefore, the value of the new centroid is $\left( {\dfrac{{11}}{3},2} \right)$.

Note: In the above question, if the triangle is reflected through the y-axis, then the sign of x-coordinate gets reversed and the sign of y-coordinate remains unchanged. But the formula remains unchanged. Also, if the triangle is reflected through the line $x = y$, then the sign of both the x-coordinate and the y-coordinate gets reversed.