The Median of a Triangle
Median of a triangle (Definition, Formula, & Examples)
The mathematical word "median" has different meanings with different operations. In statistics, it is the value lying at the midpoint of a data set.
So for a data set {3, 5, 7, 9, 11}, 7 is the median. In geometry, a median is a line segment from an interior angle of a triangle to the midpoint of the opposite side.
Learning about the geometric median can make your life in geometry, and possibly in the kitchen, easier.
Triangle
Any triangle is a polygon using three straight sides to enclose a space. Named by their angles, triangles can acute or obtuse triangles (which are grouped together as oblique triangles), or right triangles. Named by their sides, triangles can be scalene, isosceles, or equilateral triangles.
Area
Area is the space a polygon takes up in two dimensions. Every triangle has an interior space that is the triangle's area. This area is always measured in square units, no matter the shape you are measuring.
Here's the formula for area of a triangle:
What is the median of a triangle?
If you find the middle of any side of a triangle, you have found its midpoint. From that midpoint, you can construct a line segment to the opposite interior angle. That constructed line from the midpoint of a side to the opposite interior angle is a median.
Since a triangle always has three sides, it always has three medians.
Centroid of a triangle
Where medians cross, the point common to all three medians is called the centroid. The centroid is a point of concurrency. It will always be inside the triangle, unlike other points of concurrency like the orthocenter.
Medians meeting at the centroid display a peculiar property. The centroid is always two-thirds of the way along each median from that median's interior angle. This means it sets up a 2:1 ratio for each of the three medians:
2 parts of the median are between the centroid and the interior angle
1 part of the median is between the centroid and the opposite side
Center of mass
The centroid of a triangle is not just theoretical. It is the center of mass or center of gravity of the triangle. By drawing all three medians, you can find the exact spot where a physically existing triangle will balance perfectly!
This may have real-world applications for you, if you are dealing with triangles cut out of cardboard or wood. You can find the centroid and balance the triangle on a pencil point or your fingertip.
How to find the median of a triangle
A theorem, called Apollonius's Theorem, can give you the length of a median of a triangle. It is a bit wordy, but can be translated into a formula. First, the theorem:
Apollonius's Theorem states that in any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side.
As a formula, it looks like this, where a, b and c are the lengths of the sides and m is the median from interior angle A to side a:
Median of a triangle example
A median is a dividing line, separating the original triangle into two smaller triangles of equal area. That feature of a median can come in mighty handy.
Here we have △EAT, which is a scalene triangle pizza cooked up in beginning family and consumer science. The shape is not much to look at, but you made your first pizza and are proud of it. You offer to share it with a friend. How will you divide your pizza so you each get the same amount?
Use a median. From any interior angle, cut to the midpoint of the opposite side. Now you and your friend have equal amounts of pizza.
Suppose two more friends join you and want to try your oddly shaped pizza. Cut along another median, from any interior angle to the midpoint of the opposite side! The pieces are smaller and definitely strangely shaped, but they are all the same area.
Uh-oh! Two more friends want in. The pizza may look odd, but is smells wonderful. So cut the third median line, and all six of you will each enjoy the same amount of pizza, even though the shapes will all be a little…different.
Lesson summary
After working your way through this lesson, you are now able to recall that every triangle has three medians, draw or identify medians in triangles, identify the centroid of a triangle using its medians, calculate the length of a median, and relate area to medians of triangles.