### 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.

Letβs 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, letβs
take this triangle π΄πΆπ·. 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·πΆ, we would have a
line like this, line segment π΅πΈ.

Now, this theorem tells us that
this line segment π΅πΈ, 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 πΈ must be the midpoint of line segment π΄πΆ. Letβs now see how this theorem can
be proved.

Letβs begin with the same triangle
π΄πΆπ·, and we have the line segment π΅πΈ drawn from the midpoint π΅, which is
parallel to line segment πΆπ·. As we are trying to prove the
theorem, then we donβt yet know anything about the third side of line segment
π΄πΆ. We can construct a line π΄π¦ such
that π΄π¦ is parallel to the line segments π΅πΈ and π·πΆ. The reason why we do this will be
clear soon. If we consider line segments π΄π·
and π΄πΆ, 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΄πΆ 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΄πΆπ· and join the midpoints π΅ and πΈ of line segments π΄π· and π΄πΆ,
respectively, then, by the converse of the first triangle midsegment theorem, line
segments π΅πΈ and π·πΆ 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, letβs
do some investigations to prove the result. We can take this triangle πππ
and draw the line segment ππ, where π is the midpoint of line segment ππ
and π
is the midpoint of line segment ππ. We will try to work out a
relationship between the lengths of line segments ππ and π
π. To do this, letβs add the ray ππ¦
such that line segment ππ¦ is parallel to line segment ππ. Now, weβve 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 ππ is
parallel to line segment π
π.

Then, because we constructed line
segment ππ¦ parallel to line segment ππ, 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. ππ¦ is a line segment from the
midpoint π parallel to line segment ππ. 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 donβt 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.

Letβs take this one step further to
see how we can work out the length of the line segment ππ. Notice that within this triangle,
we have the quadrilateral ππππ¦. ππππ¦ 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 ππ will also be one-half π
π. Importantly, that gives us the
relationship that the length of line segment ππ 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. Weβll now see a question where we
apply the triangle midsegment theorems to find the perimeter of a shape.

Given that π· and πΈ are the
midpoints of line segments π΄π΅ and π΄πΆ, respectively, π΄π· equals 32
centimeters, π΄πΈ equals 19 centimeters, and π·πΈ equals 39 centimeters, find
the perimeter of π·π΅πΆπΈ.

We can begin by filling in the
given length information on the figure of 32 centimeters for π΄π·, 19
centimeters for π΄πΈ, and 39 centimeters for π·πΈ. We can also identify the
information that π· and πΈ are midpoints of their respective line segments. So, πΈπΆ 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΅πΆπΈ. Weβve worked out three of these
four sides, so we still need to work out the length of the line segment
πΆπ΅. To do this, letβs 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 πΆπ΅ must be parallel to line segment π·πΈ, and we know
something about the length of line segment πΆπ΅. Since π·πΈ is half the length
of πΆπ΅, we can also write that πΆπ΅ equals two times π·πΈ. We were given that π·πΈ is 39
centimeters, so doubling this, we can calculate that πΆπ΅ is 78 centimeters.

So, we now have enough
information to calculate the perimeter of π·π΅πΆπΈ. 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΅πΆπΈ is 168 centimeters.

In the next example, we will apply
the triangle midsegment theorem a number of times in the same figure. And when weβre 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 weβre working with.

In the figure shown, πΈ, πΉ,
and π· are the midpoints of line segments π΅πΆ, π΄π΅, and π΄πΆ,
respectively. Find the perimeter of triangle
πΈπΉπ·.

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. πΈ, πΉ, and π· bisect their
respective line segments. So, in order to find the
perimeter of triangle πΈπΉπ·, thatβs the distance around the outside edge, weβll
need to calculate the lengths of the three line segments πΉπ·, π·πΈ, and
πΈπΉ.

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. Letβs look at line segment
πΉπ·. Line segment πΉπ· 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΅πΆ. The length of π΅πΆ is given as
4.6 centimeters, so half of this is 2.3 centimeters.

Now, letβs see if we can do the
same to calculate the lengths of the other two sides in triangle πΈπΉπ·. We can consider the line
segment π·πΈ next. Line segment π·πΈ joins the
midpoints of two sides of a triangle, because it joins π·, the midpoint of line
segment π΄πΆ, and πΈ, the midpoint of line segment π΅πΆ. 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 πΈπΉ. It joins midpoints πΈ and
πΉ. So, the length of πΈπΉ will be
half of line segment π΄πΆ; half of 6.2 centimeters is 3.1 centimeters.

And now to find the perimeter
of triangle πΈπΉπ·, 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 πΈπΉπ· is 8.15 centimeters.

Letβs now see how we can apply the
converse of the triangle midsegment theorem to determine an unknown length.

The perimeter of square
π΄π΅πΆπ· is 352. Find π΄πΉ.

In this question, we need to
find the length of the line segment π΄πΉ, which is part of the line segment π΄π΅
on one of the sides of the square. We arenβt 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΄πΉ, we might try to guess at this point, but itβs important to
apply our geometry knowledge in questions like this so that we can prove our
calculated length is correct. So, letβs 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΄πΆ and π·π΅ are each bisected at π. But letβs focus on line segment
π΄πΆ.

By using the triangle π΄π΅πΆ,
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 ππΉ passing through the midpoint of one side of a triangle, because we
know that π is the midpoint of the line segment π΄πΆ. We also know that this same
line segment ππΉ 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΄πΉ
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.