Have you ever wondered how to find the balancing point of a triangle? The answer can be found in this lesson on the centroid of a triangle! We'll learn how medians are used to locate the centroid and then discuss its various properties.
Definition of the Centroid
The centroid of a triangle is the point where the three medians of the triangle intersect. The medians are the segments that connect a vertex to the midpoint of the opposite side. In this image, point G is the centroid of triangle ABC.
To locate the centroid of a triangle, it's easiest to draw all three medians and look for their point of intersection. To draw the median of a triangle, first locate the midpoint of one side of the triangle. Draw a line segment that connects this point to the opposite vertex. A triangle with one median drawn is shown here:
Repeat this process for the other two sides of the triangle. Once you have drawn all three medians, locate the point where all three intersect. This point is the centroid.
For a triangle made of a uniform material, the centroid is the center of gravity. Imagine that an artist designed a sculpture in which multiple metal triangles were balanced at the ends of pieces of wire. In order for the triangles to be perfectly balanced, the wires would have to be attached at the center of gravity, or centroid, of each triangle, like in the mobile shown below.
Properties of the Centroid
So what are some properties of a centroid?
The centroid is always located inside the triangle. This is true whether the triangle is acute, right, or obtuse. In this image, you can see that the centroid is inside of each the triangles, even though they all have different angle measures.
Another important property of the centroid is that it is located 2/3 of the distance from the vertex to the midpoint of the opposite side. Another way of saying this is that the centroid divides the median in a 2:1 ratio. The distance from the centroid to the vertex is twice as long as the distance from the centroid to the midpoint of the side opposite the vertex.
You can use the distance from the centroid to the vertex of a triangle to find the length of the entire median segment. You can also use this property to locate the centroid in a triangle when you can only draw one median.
So in this image, if you're told that the length of segment CF is 12 and that CF is a median of the triangle, where would the centroid, G, be located? We can use the 2/3 rule to determine how far from vertex C the centroid will be. We can use a simple equation to find the answer:
The centroid will be 8 units from vertex C, along the segment CF.
A landscape designer could use this same property if she creates a courtyard on a triangular shaped lot. She wishes to put a sculpture in the middle of the courtyard. As can be seen in this image, the three paths through the courtyard each go from a vertex to the midpoint of the opposite side. The statue will be placed at the centroid of the triangle. By using the properties of the centroid, the landscape designer can determine the distance from each vertex of the triangle to the statue.
Problems Using the Centroid
In the math classroom, nearly all problems using the centroid will also involve one or more medians in the triangle. We'll do some examples using this image as a reference:
Question: If segment CG has a length of 6 units and G is the centroid of the triangle, how long is segment CF?
Solution: Since CG is the segment joining the vertex to centroid, it is 2/3 of the total length of CF. Dividing by 2/3 will give us the length of the full segment. We can write an algebra equation and solve.
CF is 9 units long.
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Question: If segment AD has a length of 18 units and G is the centroid of the triangle, how long is segment DG?
Solution: Since G is the centroid, AD must be a median. The centroid is 2/3 of the distance along the median away from the vertex and so it is 1/3 of the distance away from point D, the midpoint of the opposite side. Again, we can write and solve the equation.
So segment DG is 6 units long.
Question: If segment EG has a length of 4 units and segments AD, BE, and CF are all medians, how long is segment BG?
Solution: In this problem, we are told the distance from a midpoint to the centroid. In this problem, we are looking for the distance from the centroid to the vertex. As noted, the centroid divides the median into a 2:1 ratio. This means that segment BG is twice the length of segment EG.
Segment BG is 8 units long.
Although each problem was slightly different, they all had in common a mention of the median or the centroid, and we were asked to find the lengths of either the entire median or a portion of the median. Keep in mind that these are the same types of calculations that the artist and the landscape architect would need to make in order to successfully locate the centroid in each of the triangles they work with!
The centroid is the point where the three medians of the triangle intersect. It has the following properties:
The centroid is always located in the interior of the triangle. The centroid is located 2/3 of the distance from the vertex along the segment that connects the vertex to the midpoint of the opposite side. The centroid is located 1/3 of the distance from the midpoint of a side along the segment that connects the midpoint to the opposite vertex. For a triangle made of a uniform material, the centroid is the center of gravity.
Lesson at a Glance
The point at which the three medians of a triangle intersect is known as the centroid. There are several properties of a centroid. It is always located in the interior of the triangle and is the center of gravity as long as the triangle is made of a uniform material.
Nearly all problems using the centroid will also involve one or more medians in the triangle.
As soon as you have a handle on this lesson, ensure that you can:
Recall and explain the properties of a centroid
Calculate the length of segments with the use of a centroid
In the following problems, students will demonstrate their knowledge of the definition of the centroid of a triangle and properties of the centroid.
1) In the following triangles, decide if the point, P, is the centroid of the triangle and support your decision with evidence.
2) Given that P is the centroid in the triangle below, find both the length of segment PD and of segment AD.
3) Is it possible for point P to be the centroid of the triangle below? Explain.
1) a. P is not the centroid of this triangle because it is not at the intersection of the medians. If it were, then the line segments extending from each vertex to the opposite side of the triangle would bisect that side of the triangle. As seen in the image, none of the three sides are bisected, so the segments are not medians, and so the point P is not the centroid.
b. P is the centroid of this triangle because it is the intersection of the three medians. It is known that the three segments which intersect to form P are the medians because each side of the triangle is bisected.
2) Since the distance from the vertex to the centroid is always 2/3 the length of the entire median, we know that AP is twice the length of PD. Since AP = 10, we have that PD = 5 and so AD = 15.
3) It is not possible for P to be the centroid of this triangle because the distance from the vertex to P is not 2/3 the length of the possible median. The distance AP = 9 and the distance PD = 5, making AD = 14 and 9 is not 2/3 of 14.
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