In this explainer, 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.

Letβs begin with understanding what the triangle midsegment theorem states.

### Theorem: Triangle Midsegment Theorem (Part 1)

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.

We can prove this by considering triangle with line segment that passes through the midpoint, , of and that is parallel to .

We can construct line such that .

Line segments and are transversals of these three parallel lines. We 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.

Given that was split into two congruent segments by the parallel lines, , , and , then the other transversal, , must also be split into two congruent segments.

Hence,

Therefore, we have proven that the third side of the triangle has been bisected.

The converse of this theorem is also true, that is, if we have a triangle where two sides are bisected by a line segment, then that line segment is parallel to the third side. This is defined below.

### Theorem: Converse of the Triangle Midsegment Theorem (Part 1)

The line segment joining the midpoints of two sides of a triangle is parallel to the third side.

We can also see one more of the triangle midsegment theorems.

### Theorem: Triangle Midsegment Theorem (Part 2)

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.

Letβs examine how we can prove this theorem. Consider , where and are the midpoints of and , respectively, such that and . We can construct ray such that and intersects at .

We can then apply the triangle midsegment theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side. This means that .

As we constructed , then by the triangle midsegment theorem, which 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, we have that is bisected. Hence, , and is the midpoint of . Furthermore, .

We can then consider quadrilateral .

is a quadrilateral with two pairs of opposite sides parallel, which by definition is a parallelogram. In a parallelogram, opposite sides are congruent. Hence, .

Therefore, we have proven the theorem: the line segment joining the midpoints of two sides of a triangle is equal to half the length of the third side.

It is worth noting that theorems 1 and 2 here are often referred to collectively as the triangle midsegment theorem, stated as follows: the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length.

In the following questions, we will see how we can apply the triangle midsegment theorem and its converse in order to find unknown side lengths, beginning with a question where we need to find the perimeter of a shape.

### Example 1: Finding the Perimeter of a Quadrilateral Using the Triangle Midsegment Theorem

Given that and are the midpoints of and , respectively, , , and , determine the perimeter of .

### Answer

We can begin by filling in the given length information on the figure, which is that , , and .

As and are the midpoints of and , we know that and

The triangle midsegment 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.

Thus, we can say that and

Given that , we have

Finally, we need to calculate the perimeter of . This is the distance around the outside of , the trapezoid on the lower part of the figure. Using , , , and , we have

Thus we have the answer that the perimeter of is 168 cm.

In the next example, we will apply the triangle midsegment theorem a number of times in different triangles in the same figure. When working through a problem such as this, it can be helpful to outline or highlight the specific triangles so that we can correctly identify the key segments that we are working with.

### Example 2: Applying the Triangle Midsegment Theorem to Solve a Problem

In the figure shown, , , and are the midpoints of , , and respectively. Find the perimeter of .

### Answer

We are given the information that , , and are the midpoints of , , and respectively. In order to calculate the perimeter of , we will need to determine the lengths of , , and .

To do this, 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.

Letβs consider .

is a line segment connecting the midpoints of two sides of a triangle. Thus, is half the length of (4.6 cm). Therefore, we have

In the same way, we can consider .

By applying the triangle midsegment theorem again, we have that must be half the length of (5.5 cm). Thus, we have

Finally, we can calculate the length of in the same way.

must be half the length of (6.2 cm). Thus, we have

Hence, as , , and , we can calculate the perimeter of as

We can give the answer that the perimeter of is 8.15 cm.

In the next example, we will see how we can apply our knowledge of the triangle midsegment theorem to help us prove geometrical properties within a given figure.

### Example 3: Completing a Proof Using the Triangle Midsegment Theorem

In the given figure, and are the midpoints of and , respectively, , and lies on . What is the shape of ?

### Answer

We are given that and are the midpoints of and respectively. Using the triangle midsegment theroem, we know that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length.

Hence, and .

We are asked to identify the shape of .

appears to be a parallelogram; however, we must prove this to be the case. A parallelogram is defined as a quadrilateral with two pairs of opposite sides parallel. We have demonstrated by the triangle midsegment theorem that , and since we are given that lies on , then .

We were given in the question that , and we proved that . Hence, is congruent to .

We have now demonstrated that has a pair of opposite sides, and , that are parallel and congruent. Hence, must be a parallelogram.

We will now see an example of how we can apply the converse of the triangle midsegment theorem to identify an unknown length.

### Example 4: Applying the Triangle Midsegment Theorem to Solve Problems given the Perimeter

The perimeter of square is 352. Find .

### Answer

In this question, we are not given any information about the lengths of any line segments. However, we are given the information that the perimeter of this square is 352 length units. Given that the perimeter is the distance around the outside edge and that a square has 4 congruent sides, we can calculate the length of one side as

At this point, we might try to *guess* the length of
; however, in questions like this,
we must apply our geometry knowledge to demonstrate and
*prove* that our calculated length is correct.

We can recall that the diagonals of a square bisect one another, so the diagonals and are bisected at . Therefore, we have that is the midpoint of .

We can then apply the converse of the triangle midsegment theorem, which 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.

Hence, is bisected by , and

We calculated earlier that the length of one side is 88 length units, so length units. Since , then is the midpoint of and . We can determine the length of as

Therefore, we can give the answer that is 44 length units.

In the final example, we will see how we can use the converse of the triangle midsegment theorem to prove a geometric property in a given figure.

### Example 5: Completing a Proof Using the Triangle Midsegment Theorem

In the given figure, which of the following is true?

- is the midpoint of .
- is the midpoint of .

### Answer

In the figure, we can observe that we have two pairs of congruent side lengths: and and and . Therefore, we can state that and are the midpoints of and respectively. We can recall that by the triangle midsegment theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length.

Thus, in the figure, we have that .

We will now consider the choices that we are presented with and determine which of them is true.

In choice A, we address the statement that is the midpoint of . In fact, we have already proven that is the midpoint of . We must be careful not to confuse the two line segments. Here, we cannot prove that is the midpoint of .

Next, letβs see the statement in choice : is the midpoint of . We can consider triangle .

Given that passes though the midpoint of and is parallel to , then by the converse of the triangle midsegment theorem, it must bisect the third side, . Hence, , and is the midpoint of . The statement given in choice B is true.

In choice C, we need to determine if the statement that is true. Letβs consider . Applying the triangle midsegment theorem here would allow us to prove that .

Since can be observed not to lie on point and , then cannot also be the same length as . Thus, this statement is not true.

Finally, letβs consider choice D: . We cannot apply the triangle midsegment theorem or its converse to demonstrate that this statement is true. As with choice C, the line segment that can be demonstrated to be half the length of is , not .

Therefore, the true statement is given in choice B: is the midpoint of .

We have seen how we can apply the triangle midsegment theorem and its converse. We can now summarize the key points.

### Key Points

- 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 line segment joining the midpoints of two sides of a triangle is parallel to the third side.
- 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.