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

Question

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

Aaliyah Anderson · Accepted Answer

New centroid #color(indigo)(G’(x, y) => ( 11/3, -2)#

Reflection rules

Reflection about the x - axis #color(red)(x,y) color(blue)(-> )color(purple)(x, -y)#

Three points A, B, C will become A’, B’, C’.

#color(red)(A ( 1, 3) ) color(blue)(- >) color(purple)( A’ (1, -3))#

#color(red)(B ( 2, -4) ) color(blue)(- >) color(purple)( B’ (2, 4))#

#color(red)(A ( 8, -5) ) color(blue)(- >) color(purple)( C’ (8, 5))#

New Centroid is found out using the formula

#color(green)G’_x = (x_A + x_B + x_C) / 3 = (1 + 2 + 8) / 3 = color(green)(11/3)#

#color(green)G’_y = (y_A + y_B + y_C) / 3 = (-3 + 4 + 5) / 3 = color(green)(2)#

New centroid #color(indigo)(G’(x, y) => ( 11/3, -2)#

Sadie Allen · Answer

undefined The centroid of a triangle is the point where the three medians intersect. When a triangle is reflected across the x-axis, the x-coordinates of its vertices remain the same, while the y-coordinates are negated.

Given the original vertices (1, 3), (2, -4), and (8, -5), after reflection across the x-axis, they become (1, -3), (2, 4), and (8, 5) respectively.

To find the centroid of the reflected triangle, we calculate the average of the x-coordinates and the average of the y-coordinates of its vertices.

The new centroid coordinates will be:
$$ x = \frac{1 + 2 + 8}{3} = \frac{11}{3} $$
$$ y = \frac{(-3) + 4 + 5}{3} = \frac{6}{3} = 2 $$

Therefore, the new centroid of the reflected triangle is ($\frac{11}{3}, 2$).