IXL - Find the centroid of a triangle (Geometry practice)

What are the coordinates of the centroid of

△

Simplify any fractions.

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Learn with an example

question

What are the coordinates of the centroid of

△

Simplify any fractions.

key idea

The point where a triangle's three medians intersect is called the

centroid.

A median of a triangle is a segment connecting a vertex to the midpoint of its opposite side.

solution

To find the centroid of

△

you can find the point at which two of its medians intersect. So, find the equations of the lines containing these medians, and then solve the system of equations.

Find the equation of the line containing the median from point

The line containing this median passes through

(

)

and the midpoint of

. To find the midpoint between

(

–

)

and

(

)

, use the midpoint formula.

midpoint

–

(

)

Now use the slope formula to find the slope of the line passing through

(

)

and the midpoint of

(

)

slope

–

Plug the slope and the point

(

)

into the point-slope formula to find the equation of the line in slope-intercept form.

–

(

–

)

–

(

–

)

–

Find the equation of the line containing the median from point

The line containing this median passes through

(

–

)

and the midpoint of

. To find the midpoint between

(

)

and

(

)

, use the midpoint formula.

midpoint

(

)

Now use the slope formula to find the slope of the line passing through

(

–

)

and the midpoint of

(

)

slope

–

(

–

)

–

Plug the slope and the point

(

–

)

into the point-slope formula to find the equation of the line in slope-intercept form.

–

(

–

)

–

(

–

)

(

–

)

–

Solve the system of equations.

You now have a system of two linear equations:

y
=
–
3
2
x
+
15
2
y
=
9
x
–
31

To solve the system, you can set the right sides of both equations equal to each other since they are both equal to

Then, solve for

–

(

–

)

–

Now, plug

into the second equation to find

–

The solution to the system of equations is

This solution is the point at which the lines intersect, and so it is also the point where the medians of the triangle intersect. That means

is the centroid of

△

This diagram shows

△

its three medians, and the centroid.

There is another way to find the centroid of

△

The centroid is the center of mass, or balance point, of a triangle. So, to find the centroid of

△

you could average the

and

y-values

of vertices

(

)

(

–

)

and

(

)

(

–

)

Looking for more? Try this:

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Teacher tools

Work it out

Not feeling ready yet? These can help:

N.3Identify medians, altitudes, angle bisectors, and perpendicular bisectors

B.10Midpoint formula: find the midpoint

E.3Equations of lines

O.14Solve a system of equations using any method

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