### Video Transcript

Calculate the perimeter of
π·πΈπΆπΉ.

In the figure, we can observe that
we have a large triangle on the outside, triangle π΄π΅πΆ. And on the sides of this triangle,
we have three pairs of congruent line segments marked. Firstly, line segments πΆπΈ and
π΅πΈ are congruent. Line segments π΄π· and π΅π· are
congruent. And line segments π΄πΉ and πΆπΉ are
congruent. And so we could describe points π·,
πΈ, and πΉ as midpoints of their respective sides in triangle π΄π΅πΆ.

Now, given that we have to find the
perimeter of π·πΈπΆπΉ, which is the shaded polygon in the figure, then knowing that
we have the midpoints of the sides of the triangle will be very useful. Because we can apply the triangle
midsegment theorem, which is 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.

So letβs firstly consider line
segment πΉπ·, which is a line connecting the midpoints of two sides of the
triangle. By the theorem, we know that it
must be parallel to the third side, which is the line segment πΆπ΅, and half the
length of line segment πΆπ΅. Since we are given on the diagram
that line segment πΆπ΅ has a length of 159.6 centimeters, then halving this, we can
calculate that line segment πΉπ· has a length of 79.8 centimeters.

Now letβs think about the line
segment π·πΈ. By applying the theorem once more,
we know that this line segment must be parallel to the third side π΄πΆ and half its
length. So we can write that π·πΈ equals
one-half π΄πΆ. Given the measurement of 142.4
centimeters for line segment π΄πΆ, then halving this gives us that the line segment
π·πΈ has a length of 71.2 centimeters.

Now, we can return to the fact that
we need to find the perimeter of π·πΈπΆπΉ, which is the distance around the outside
edge. We have worked out the length of
two of the sides of this quadrilateral, but what about the other two sides? Well, we can consider that we have
worked out that π·πΈπΆπΉ has two pairs of opposite sides parallel. And so, by definition, π·πΈπΆπΉ is
a parallelogram. And one of the properties of
parallelograms is that opposite sides are congruent. So we know that line segment πΆπΈ
must also be 79.8 centimeters. And line segment πΆπΉ is 71.2
centimeters.

Therefore, to find the perimeter,
we add the lengths of the four sides of 79.8, 71.2, 79.8, and 71.2 centimeters,
which gives us an answer for the perimeter of π·πΈπΆπΉ as 302 centimeters.