Question

A triangle has corners at `(4, 6 )`, ( 1 , 7)`, and`( 3, -4)#. If the triangle is reflected across the x-axis, what will its new centroid be?

Answer 1

`(8/3,-3)`

Explanation:

The first step here is to find the coordinates of the centroid of the triangle.

Given the 3 vertices `(x_1,y_1),(x_2,y_2)" and " (x_3,y_3)`

Then the coords of the centroid are found as follows.

`color(red)(|bar(ul(color(white)(a/a)color(black)(1/3(x_1+x_2+x_3),1/3(y_1+y_2+y_3))color(white)(a/a)|)))`

here let `(x_1,y_1)=(4,6),(x_2,y_2)=(1,7),(x_3,y_3)=(3,-4)`

x-coord `=1/3(4+1+3)=8/3"`

and y-coord `=1/3(6+7-4)=3`

coords of centroid `=(8/3,3)`

now, under a reflection in the x-axis

a point (x ,y) → (x ,-y)

hence coords of centroid `(8/3,3)rArr(8/3,-3)`

Answer 2

New centroid is `(8/3,-3)`

Explanation:

The centroid of the triangle with corners at `(4,6)`, `(1,7)` and `(3, -4)` is

`((4+1+3)/3,(6+7+(-4))/3)` or `(8/3,3)`

When triangle is reflected across the x-axis, its centroid too is reflected across the x-axis

and as reflection of a point `(x_1,y_1)` in x-axis is `(x_1,-y_1)`,

New centroid is `(8/3,-3)`