# A triangle has corners at #(1, 6 )#, ( 1 , 2)#, and #( 7, -1)#. If the triangle is reflected across the x-axis, what will its new centroid be?

##### 2 Answers

#### Explanation:

If a point

For information on centroids, visit this webpage.

Now, if we can find out the three medians of the given triangle, and find their point of intersection, we will have the centroid of the triangle. From there, we only need to translate this point from

The process of finding out the medians of a triangle is quite tedious, so I will only explain how to find out the first. But don't worry, it will become apparent from this one example how to figure out the other two.

We will now figure out the median that originates at point

where

Applying this formula to the points

Now to return to finding the median of the first point of our triangle. We must now write a linear equation for a line that passes through the points

Solving for the system of linear equations

by means of substitution, we obtain:

Therefore the linear equation that describes our first median can be written

After finding out the remaining two medians, we set them all equal to find for which

Simplifying and solving, we obtain

The centroid of the triangle given is therefore

#### Explanation:

If the given triangle is reflected across the x axis, then the centroid will also be reflected in the x axis.

Let the triangle be

The co-ordinates of the centroid of a triangle are the arithmetic mean of the x and y co-ordinates of the vertices.

Centroid is:

Centroid of

A reflection in the x axis maps:

So image of centroid is: