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Lesson Video: Medians of Triangles Mathematics • 11th Grade

In this video, we will learn how to identify the medians of a triangle and use their properties of proportionality to find a missing length.

14:45

Video Transcript

In this video, we will learn how to identify medians of a triangle and, using their properties of proportionality, find a missing length. First of all, let’s think about what a median is in a triangle. The medians of a triangle are the three segments going from each vertex to the midpoint of the opposite side. Let’s see if we can visualize this.

If we start with a triangle, the medians are the line segments going from the vertex to the midpoint of the opposite side. The midpoint of the opposite side will be the point that divides that line segment in half. The midpoint is here as it divides this segment in half, and that makes this our first median. For our second vertex, we find the midpoint of the opposite side, and then we draw a line from the vertex to that midpoint. And we have a second median in this triangle. And here is the third median.

And at this point, you might notice something about these three medians. The medians of this triangle intersect at one point. And this is true for every triangle; the three medians intersect at a single point. They are concurrent. The point of concurrency is where the three lines intersect, and this point has a special name. It’s called the centroid. The centroid is always located inside the triangle.

We’ll now consider some properties that we find with these medians and the centroid. We have the centroid theorem, which tells us the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. Let’s try to visualize this. In this triangle, the three medians have been divided into thirds, and the centroid is 𝑃. The distance from each vertex to the centroid is two-thirds. And the remaining one-third is the distance between the centroid and the midpoint. That means the line segment 𝐴𝑃 is two-thirds of the median, and the line segment 𝑃𝐸 is one-third of the median.

We can see here that the line segment 𝐴𝑃 is twice the line segment of 𝑃𝐸. Let’s try to write this algebraically. Line segment 𝐴𝑃 is equal to two-thirds of the whole median 𝐴𝐸, and line segment 𝑃𝐸 is equal to one-third of the whole median 𝐴𝐸. If we take our second equation and we multiply through by three, we see that three times 𝑃𝐸 equals 𝐴𝐸.

Since we found that line segment 𝐴𝐸 equals three times line segment 𝑃𝐸, in our first equation, we can substitute our new value in for line segment 𝐴𝐸. If line segment 𝐴𝑃 is equal to two-thirds line segment 𝐴𝐸 and line segment 𝐴𝐸 equals three times line segment 𝑃𝐸, two-thirds times three equals two. And we’ve confirmed what we intuitively saw was true, that line segment 𝐴𝑃 is equal to two times line segment 𝑃𝐸. Likewise, it would be fair to say that line segment 𝑃𝐸 equals half of line segment 𝐴𝑃, as one-third times two equals two-thirds and two-thirds times one-half equals one-third. Using this information, let’s consider some examples.

In a triangle 𝐴𝐵𝐶, 𝑀 is the point of concurrency of its medians. If line segment 𝐴𝐷 is a median, then 𝐴𝑀 is equal to blank of 𝑀𝐷.

First of all, we know that the point of concurrency of its medians in a triangle is its centroid. If we wanted to sketch a triangle to try and understand what’s happening here, we would need triangle 𝐴𝐵𝐶 and then we could sketch a centroid. We know that the point of concurrency, the centroid, is point 𝑀 and that line 𝐴𝐷 is a median. The centroid theorem tells us that the distance from the vertex to the centroid is two-thirds of the median, and the distance from the centroid to the midpoint is one-third of the median. And we want to compare the relationship between 𝑀𝐷 and 𝐴𝑀. To go from 𝑀𝐷 to 𝐴𝑀, to get from one-third to two-thirds, we multiply by two. 𝐴𝑀 is twice 𝑀𝐷, which means we would find 𝐴𝑀 by multiplying 𝑀𝐷 by two.

In our next example, we’ll use the properties of medians to find a missing side length.

What is the length of line segment 𝐶𝐷?

On this diagram, we can go ahead and identify line segment 𝐶𝐷 as here. From there, it might be helpful to write out what we know based on this figure. We know that line segment 𝐵𝐸 is equal to line segment 𝐸𝐶. We also know that line segment 𝐵𝐹 is equal to line segment 𝐹𝐴. We know that line segment 𝐵𝐷, 𝐴𝐸, and 𝐶𝐹 all intersect at one point. And we know the measure of line segment 𝐴𝐶 is 13.8 centimeters.

What can we say based on this information? Because the point 𝐸 divides line 𝐵𝐶 in half, this is a midpoint, and the distance from a vertex to a midpoint is a median. That means we can say that line segment 𝐴𝐸 is a median. For that same reason, point 𝐹 is a midpoint, and so we can say line segment 𝐶𝐹 is a median because 𝐴𝐸 and 𝐶𝐹 are medians and line segment 𝐵𝐷 intersects at the same point as the other two medians. They have a point of concurrency, which makes line segment 𝐵𝐷 also a median and tells us that line segment 𝐴𝐷 will be equal in length to line segment 𝐷𝐶.

Then we know to find the value of line segment 𝐶𝐷, it will be equal to half of 13.8. And 13.8 times one-half is 6.9. And so we say that the line segment 𝐶𝐷 has a length of 6.9 centimeters.

In our next example, we’re given the length of a median and we’ll need to use that value to find the length of the distance from a vertex to the centroid.

Find the length of line segment 𝐴𝑀, given that 𝐴𝐸 equals 54.

Let’s see what we can tell from the diagram. The point 𝐷 divides line segment 𝐴𝐵 in half, and the point 𝐸 divides line segment 𝐵𝐸 in half. So we have two midpoints. And we know that 𝐴 and 𝐶 are vertices of this triangle, which means that line segment 𝐴𝐸 and line segment 𝐶𝐷 are medians of this triangle. The place where medians intersect inside a triangle is called the point of concurrency, or the centroid. And we know based on the centroid theorem that the distance from the vertex to the centroid is two-thirds of the median, and the distance from the centroid to the midpoint is one-third of the median.

This means line segment 𝐴𝑀 is equal to two-thirds the median 𝐴𝐸. And since we know 𝐴𝐸, the median, equals 54, we can say that the length of line segment 𝐴𝑀 will be equal to two-thirds of 54. If we wanna simplify this, I know that 54 divided by three equals 18 and two times 18 equals 36. So we can say that line segment 𝐴𝑀 measures 36.

In our next example, we’re given the distance from a vertex to a centroid, and we need to find the distance from a centroid to a midpoint.

In triangle 𝐽𝐾𝐿, 𝐽𝑃 equals six centimeters. Find the length of line segment 𝑃𝑆.

We need to know the length of line segment 𝑃𝑆, and we’ve been given that 𝐽𝑃 equals six centimeters. First of all, we should note that the point 𝑆 divides the line segment 𝐾𝐿 in half, which makes 𝑆 a midpoint. Since 𝐽 is a vertex of this triangle, we know that the distance between a vertex and the midpoint is called the median. And so we can say that 𝐽𝑆 is a median. In the same way, 𝑇 divides line segment 𝐽𝐿 and 𝑅 divides line segment 𝐽𝐾, which means 𝐾𝑇 is a median and 𝐿𝑅 is a median. Since 𝐽𝑆, 𝐾𝑇, and 𝐿𝑅 are all meet at one point 𝑃, 𝑃 is the point of concurrency or the centroid.

This is important because we know something about the centroid. For a median, the distance between the vertex and the centroid is two-thirds of the median. And the distance between the centroid and the midpoint is one-third of the distance of that median. To go from two-thirds to one-third, we multiply by one-half. One-third is half of two-thirds. And so we can say that the line segment 𝑃𝑆 will be equal to one-half the line segment 𝐽𝑃. Since 𝐽𝑃 was equal to six centimeters, we take one half of that and we say that the line segment 𝑃𝑆 is equal to three centimeters.

In our final example, we’ll again be given the distance from a vertex to a centroid. And we’ll try to solve for a missing variable in the distance from the centroid to the midpoint.

In triangle 𝐾𝑀𝐻, 𝐾𝑄 equals two and 𝑄𝑃 equals five 𝑥 minus seven. Find 𝑥.

First, we want to look at our diagram and see what we know. We see that the points 𝐽, 𝑃, and 𝐿 divide each side of the triangles in half. And then we have lines from each of the vertices to those points. The distance from a vertex to a midpoint is the median. And that means 𝐻𝐿, 𝑀𝐽, and 𝐾𝑃 are all medians of this triangle. And we know the point of concurrency for three medians is the centroid. We can also write down some other information we know, that 𝐾𝑄 is two and that 𝑄𝑃 is five 𝑥 minus seven. Because point 𝑄 is the centroid, 𝐾𝑄 is equal to two-thirds of 𝐾𝑃. That is, the distance from the vertex to the centroid is two-thirds of the distance of the median. And then 𝑄𝑃 equals one-third of the distance of 𝐾𝑃.

And so we can say that 𝐾𝑄 is equal to two times 𝑄𝑃. Or we can say that 𝑄𝑃 is equal to one-half of 𝐾𝑄 because two-thirds is equal to one-third times two or one-third is equal to two-thirds times one-half. Using the formula 𝑄𝑃 is equal to one-half times 𝐾𝑄, we plug in the values we know for 𝑄𝑃 and 𝐾𝑄. And we get five 𝑥 minus seven equals one-half times two. One-half times two is one, so we have five 𝑥 minus seven equals one. And we add seven to both sides. Five 𝑥 equals eight, and 𝑥 will equal eight divided by five, which is 1.6.

At this point, it’s probably worth plugging 𝑥 back in to our five 𝑥 minus seven to make sure that this answer seems reasonable. We know that 𝑄𝑃 will be equal to five times 𝑥 minus seven. Five times 1.6 is eight; eight minus seven equals 𝑄𝑃. And that means 𝑄𝑃 is equal to one. It is true that one is half of two and that two is one times two. This confirms the proportionality of our median and confirms that 𝑥 equals 1.6.

Before we finish, let’s review our key points. The medians of a triangle are the three segments going from each vertex to the midpoint of the opposite side. The three medians intersect at a single point. This point of concurrency is called the centroid. The centroid theorem states that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. This means that the segment between the vertex and the centroid is twice the segment between the centroid and the midpoint of the opposite side.

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