A triangle has corners at #(8, 3 )#, ( 5, -8)#, and #(7, -4 )#. If the triangle is reflected across the x-axis, what will its new centroid be?

Question

A triangle has corners at #(8, 3  )#, ( 5,  -8)#, and #(7,  -4  )#.  If the triangle is reflected across the x-axis, what will its new centroid be?

Violet Aldrich · Accepted Answer

#(20/3,3)#

The first step is to find the coordinates of the centroid, #(x_c,y_c)#

Given that the vertices of a triangle are #(x_1,y_1),(x_2,y_2),(x_3,y_3)#

Then.

#x_c=1/3(x_1+x_2+x_3)" the average of the x-coordinates"#

and #y_c=1/3(y_1+y_2+y_3)" the average of the y-coordinates"#

Here.

#(x_1,y_1)=(8,3),(x_2,y_2)=(5,-8), (x_3,y_3)=(7,-4)#

#rArrx_c=1/3(8+5+7)=20/3#

and #y_c=1/3(3-8-4)=-3#

coordinates of centroid #=(20/3,-3)#

Under reflection in the x-axis, a point (x ,y) → (x ,-y)

#rArr(20/3,-3)to(20/3,3)#

Maverick Smith · Answer

undefined To find the centroid of the reflected triangle across the x-axis, you would need to reflect each of the original vertices across the x-axis to obtain their new coordinates. Then, you would calculate the average of the x-coordinates and the average of the y-coordinates of the new vertices to find the centroid.

For each vertex (x, y), its reflection across the x-axis would have coordinates (x, -y).

The original vertices are:
A(8, 3), B(5, -8), and C(7, -4).

The reflected vertices would be:
A'(8, -3), B'(5, 8), and C'(7, 4).

Now, to find the centroid, calculate the average of the x-coordinates and the average of the y-coordinates of the reflected vertices:

Average of x-coordinates:
$$ \frac{8 + 5 + 7}{3} = \frac{20}{3} $$

Average of y-coordinates:
$$ \frac{-3 + 8 + 4}{3} = \frac{9}{3} = 3 $$

Therefore, the coordinates of the centroid of the reflected triangle across the x-axis are $\left(\frac{20}{3}, 3\right)$.