A triangle has corners at #(8, 1 )#, ( 5, -8)#, and #(7, -2 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Question

A triangle has corners at #(8, 1  )#, ( 5,  -8)#, and #(7,  -2  )#.  If the triangle is reflected across the x-axis, what will its new centroid be?

Aaliyah Anderson · Accepted Answer

The coordinates of the centroid of the triangle after it has been reflected over the #x# axis are #(20/3,3)#.

The reflection of #(x,y)# over the #x# axis is #(x, -y)#. If a triangle has vertices of #(8,1), (5,-8), (7,-2)#, the vertices after reflecting it over the #x# axis are #(8,-1), (5,8), (7,2)#. To find the coordinates of a centroid of a triangle with vertices at #(x_1, y_1), (x_2,y_2), (x_3,y_3)#, use the formula #(x,y)_"centroid"= ((x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3)# #(x,y)_"centroid"=((8+5+7)/3, (-1+8+2)/3)=(20/3,3)#

Sophia Alfred · Answer

undefined To find the centroid of a triangle after it's reflected across the x-axis, we need to find the coordinates of the original triangle's centroid and then reflect it across the x-axis.

The centroid of a triangle is the point of intersection of its medians. Each median divides the triangle into two equal areas. The coordinates of the centroid are found by averaging the coordinates of the triangle's vertices.

Given the vertices of the original triangle are (8, 1), (5, -8), and (7, -2), we calculate the average of the x-coordinates and the average of the y-coordinates to find the centroid.

The centroid coordinates are:

$x_c = \frac{8 + 5 + 7}{3} = \frac{20}{3} $

$y_c = \frac{1 - 8 - 2}{3} = \frac{-9}{3} = -3 $

Now, when the triangle is reflected across the x-axis, the x-coordinate remains the same, but the sign of the y-coordinate changes.

So, the new centroid will have the coordinates:

$x_c' = \frac{20}{3} $

$y_c' = -(-3) = 3 $

Therefore, the new centroid of the reflected triangle will be at the point $ \left(\frac{20}{3}, 3\right) $.