# Lesson Video: Triangle Midsegment Theorems Mathematics

In this video, we will learn how to use the triangle midsegment theorem to prove the parallelism of lines in a triangle or find a missing side length.

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### Video Transcript

In this video, we will learn how to use the triangle midsegment theorem to prove that lines are parallel or find a missing side length in a particular triangle scenario.

Letβs begin by looking at the first of the triangle midsegment theorems. This theorem states that the line segment passing through the midpoint of one side of a triangle that is also parallel to another side of a triangle bisects the third side of the triangle. To see this on a diagram, letβs take this triangle π΄πΆπ·. Taking the midpoint of one of the sides, so using the midpoint π΅ of line segment π΄π·, and then drawing a line parallel to the base, which we can take as the line segment π·πΆ, we would have a line like this, line segment π΅πΈ.

Now, this theorem tells us that this line segment π΅πΈ, which is from the midpoint of one of the sides of the triangle and parallel to another side, is then a bisector of the third side. Equivalently, that means that this point πΈ must be the midpoint of line segment π΄πΆ. Letβs now see how this theorem can be proved.

Letβs begin with the same triangle π΄πΆπ·, and we have the line segment π΅πΈ drawn from the midpoint π΅, which is parallel to line segment πΆπ·. As we are trying to prove the theorem, then we donβt yet know anything about the third side of line segment π΄πΆ. We can construct a line π΄π¦ such that π΄π¦ is parallel to the line segments π΅πΈ and π·πΆ. The reason why we do this will be clear soon. If we consider line segments π΄π· and π΄πΆ, these are both transversals of the three parallel lines.

Recall that if a set of parallel lines divides a transversal into segments of equal lengths, then that set divides any other transversal into segments of equal lengths. So, because line segment π΄π· was split into two congruent pieces, we know that line segment π΄πΆ will be the same. It will also be split into two congruent pieces. In other words, this third side of the triangle has been bisected, thus proving this triangle midsegment theorem.

Now, the converse of this theorem also holds true. We can state this as the line segment joining the midpoints of two sides of a triangle is parallel to its third side. So, this time, if we take triangle π΄πΆπ· and join the midpoints π΅ and πΈ of line segments π΄π· and π΄πΆ, respectively, then, by the converse of the first triangle midsegment theorem, line segments π΅πΈ and π·πΆ are parallel. We will now see the final triangle midsegment theorem. The second part of the triangle midsegment theorem is all about the lengths of two important line segments.

Before we state this theorem, letβs do some investigations to prove the result. We can take this triangle πππ and draw the line segment ππ, where π is the midpoint of line segment ππ and π is the midpoint of line segment ππ. We will try to work out a relationship between the lengths of line segments ππ and ππ. To do this, letβs add the ray ππ¦ such that line segment ππ¦ is parallel to line segment ππ. Now, weβve already seen the property that the line segment joining the midpoints of two sides of a triangle is parallel to the third side; that was the converse of the first part of the theorem. Therefore, the line segment ππ is parallel to line segment ππ.

Then, because we constructed line segment ππ¦ parallel to line segment ππ, we can apply the first triangle midsegment theorem. The line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle. ππ¦ is a line segment from the midpoint π parallel to line segment ππ. And that means that line segment ππ is bisected. This may a little harder to visualize at first, but it also illustrates the point that we donβt always need to have the parallel sides as horizontal with the bisected sides upwards from that. And this means that we now know that each of the line segments ππ¦ and π¦π will be equal to half the length of the whole line segment ππ.

Letβs take this one step further to see how we can work out the length of the line segment ππ. Notice that within this triangle, we have the quadrilateral ππππ¦. ππππ¦ has two pairs of parallel sides, and so, by definition, it is a parallelogram. Parallelograms have opposite sides congruent. Therefore, the length of line segment ππ will also be one-half ππ. Importantly, that gives us the relationship that the length of line segment ππ is half the length of line segment ππ. We can now formally define the second part of the triangle midsegment theorem. The length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side.

Before we look at some questions, we can note that sometimes what we saw as the converse of part one and part two of the triangle midsegment theorems are combined into one theorem. This is often referred to collectively as the triangle midsegment theorem, stated as the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. Weβll now see a question where we apply the triangle midsegment theorems to find the perimeter of a shape.

Given that π· and πΈ are the midpoints of line segments π΄π΅ and π΄πΆ, respectively, π΄π· equals 32 centimeters, π΄πΈ equals 19 centimeters, and π·πΈ equals 39 centimeters, find the perimeter of π·π΅πΆπΈ.

We can begin by filling in the given length information on the figure of 32 centimeters for π΄π·, 19 centimeters for π΄πΈ, and 39 centimeters for π·πΈ. We can also identify the information that π· and πΈ are midpoints of their respective line segments. So, πΈπΆ also has a length of 19 centimeters, and π·π΅ has a length of 32 centimeters. We know that we need to work out the perimeter of this quadrilateral π·π΅πΆπΈ. Weβve worked out three of these four sides, so we still need to work out the length of the line segment πΆπ΅. To do this, letβs use the triangle midsegment theorem.

This theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. This means we can recognize that line segment πΆπ΅ must be parallel to line segment π·πΈ, and we know something about the length of line segment πΆπ΅. Since π·πΈ is half the length of πΆπ΅, we can also write that πΆπ΅ equals two times π·πΈ. We were given that π·πΈ is 39 centimeters, so doubling this, we can calculate that πΆπ΅ is 78 centimeters.

So, we now have enough information to calculate the perimeter of π·π΅πΆπΈ. Recall that the perimeter is the distance around the outside edge of a shape. So that means we need to add the four lengths of 39, 32, 78, and 19 centimeters, which gives us the final answer that the perimeter of π·π΅πΆπΈ is 168 centimeters.

In the next example, we will apply the triangle midsegment theorem a number of times in the same figure. And when weβre working through a problem like this, it can be really helpful to highlight the specific line segments so that we can easily identify the key parts that weβre working with.

In the figure shown, πΈ, πΉ, and π· are the midpoints of line segments π΅πΆ, π΄π΅, and π΄πΆ, respectively. Find the perimeter of triangle πΈπΉπ·.

We should note that from the information we are given and the markings on the diagram, that we have three midpoints of line segments here. πΈ, πΉ, and π· bisect their respective line segments. So, in order to find the perimeter of triangle πΈπΉπ·, thatβs the distance around the outside edge, weβll need to calculate the lengths of the three line segments πΉπ·, π·πΈ, and πΈπΉ.

Now, because we know that there are some midpoints of lines, that might make us wonder if we could possibly apply one of the triangle midsegment theorems. We can recall that the length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side. Letβs look at line segment πΉπ·. Line segment πΉπ· is a line segment joining the midpoints of two sides of a triangle. Therefore, its length is going to be half the length of the third side, which is line segment π΅πΆ. The length of π΅πΆ is given as 4.6 centimeters, so half of this is 2.3 centimeters.

Now, letβs see if we can do the same to calculate the lengths of the other two sides in triangle πΈπΉπ·. We can consider the line segment π·πΈ next. Line segment π·πΈ joins the midpoints of two sides of a triangle, because it joins π·, the midpoint of line segment π΄πΆ, and πΈ, the midpoint of line segment π΅πΆ. Therefore, we know that it must be half the length of line segment π΄π΅, which is the third side of the triangle. Half of 5.5 centimeters is 2.75 centimeters. And we can do the same for the length of line segment πΈπΉ. It joins midpoints πΈ and πΉ. So, the length of πΈπΉ will be half of line segment π΄πΆ; half of 6.2 centimeters is 3.1 centimeters.

And now to find the perimeter of triangle πΈπΉπ·, we add these three calculated lengths together. 2.3 plus 2.75 plus 3.1 is equal to 8.15 centimeters. And so, by applying the triangle midsegment theorem three times, we have determined that the perimeter of triangle πΈπΉπ· is 8.15 centimeters.

Letβs now see how we can apply the converse of the triangle midsegment theorem to determine an unknown length.

The perimeter of square π΄π΅πΆπ· is 352. Find π΄πΉ.

In this question, we need to find the length of the line segment π΄πΉ, which is part of the line segment π΄π΅ on one of the sides of the square. We arenβt given any length information on the diagram, but we are told that the perimeter of the square is 352 length units.

Now, given that the perimeter is the distance around the outside edge of a shape, and we know that the shape is a square, then we can work out one of the side lengths. Because four times the length of one side would give us the perimeter of a square, to work out the length of one side of this square, we divide the perimeter of 352 by four. That gives us 88 length units. So, all the sides of the square will have a length of 88 length units.

Now, given that we want to find the length of π΄πΉ, we might try to guess at this point, but itβs important to apply our geometry knowledge in questions like this so that we can prove our calculated length is correct. So, letβs think about what we know of the geometry of a square. In particular, we can recall that in a square, the diagonals bisect one another. So, the diagonals of line segments π΄πΆ and π·π΅ are each bisected at π. But letβs focus on line segment π΄πΆ.

By using the triangle π΄π΅πΆ, we observe that we can apply one of the triangle midsegment theorems here. This theorem states that the line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle. Here, we have got the line segment ππΉ passing through the midpoint of one side of a triangle, because we know that π is the midpoint of the line segment π΄πΆ. We also know that this same line segment ππΉ is parallel to another side of the triangle because of the markings on the diagram. And so, that means that the third side of the triangle must be bisected.

The line segment π΄π΅, which we calculated has a length of 88 length units, is split into two equal pieces. And 88 divided by two equals 44. And so, by applying this triangle midsegment theorem, we have found that the length of line segment π΄πΉ is 44 length units.

We can now summarize the key points of this video. We saw, and proved, the first part of the triangle midsegment theorem. The line segment passing through the midpoint of one side of a triangle that is also parallel to another side of the triangle bisects the third side of the triangle.

The converse of the first theorem is also true. That is, the line segment joining the midpoints of two sides of a triangle is parallel to the third side. We also saw that the length of the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side.

The second and third theorems listed here may often be combined in the theorem that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. And so, as displayed in this diagram, we can see that these theorems give us a relationship between the midpoints of sides of a triangle and the parallelism and lengths of a midsegment and a side of the triangle.