Solveeit Logo
Login

Question

Question: A triangle has corners at \(\left( {1,3} \right),\,\left( {2, - 4} \right)\,and\,\left( {8, - 5} \ri...

A triangle has corners at (1,3), (2,−4) and (8,−5)\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?

Explanation

Solution

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=x1+x2+x33x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3} and y=y1+y2+y33y = \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 (1,3), (2,−4) and (8,−5)\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 (1,−3), (2,4) and (8,5)\left( {1, - 3} \right),\,\left( {2,4} \right)\,and\,\left( {8,5} \right).

Now, put the values x1=1 , x2= 2 , x3=8  and y1=−3 , y2 = 4 , y3= 5{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=x1+x2+x33x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3} and y=y1+y2+y33y = \dfrac{{{y_1} + {y_2} + {y_3}}}{3}, we get
⇒x=x1+x2+x33\Rightarrow x = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}
∴x=1+2+83\therefore x = \dfrac{{1 + 2 + 8}}{3}
On simplification, we get
⇒x=113\Rightarrow x = \dfrac{{11}}{3}
Now, we will find the y-coordinate
⇒y=y1+y2+y33\Rightarrow y = \dfrac{{{y_1} + {y_2} + {y_3}}}{3}
⇒y=−3+4+53\Rightarrow y = \dfrac{{ - 3 + 4 + 5}}{3}
⇒y=63\Rightarrow y = \dfrac{6}{3}
∴y=2\therefore y = 2

Therefore, the value of the new centroid is (113,2)\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=yx = y, then the sign of both the x-coordinate and the y-coordinate gets reversed.