# Question Video: Finding the Perimeter of a Quadrilateral Using the Triangle Midsegment Theorem Mathematics

Calculate the perimeter of π·πΈπΆπΉ.

03:52

### Video Transcript

Calculate the perimeter of π·πΈπΆπΉ.

In the figure, we can observe that we have a large triangle on the outside, triangle π΄π΅πΆ. And on the sides of this triangle, we have three pairs of congruent line segments marked. Firstly, line segments πΆπΈ and π΅πΈ are congruent. Line segments π΄π· and π΅π· are congruent. And line segments π΄πΉ and πΆπΉ are congruent. And so we could describe points π·, πΈ, and πΉ as midpoints of their respective sides in triangle π΄π΅πΆ.

Now, given that we have to find the perimeter of π·πΈπΆπΉ, 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 letβs firstly consider line segment πΉπ·, 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 πΆπ΅, and half the length of line segment πΆπ΅. Since we are given on the diagram that line segment πΆπ΅ has a length of 159.6 centimeters, then halving this, we can calculate that line segment πΉπ· has a length of 79.8 centimeters.

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

Now, we can return to the fact that we need to find the perimeter of π·πΈπΆπΉ, 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·πΈπΆπΉ has two pairs of opposite sides parallel. And so, by definition, π·πΈπΆπΉ is a parallelogram. And one of the properties of parallelograms is that opposite sides are congruent. So we know that line segment πΆπΈ must also be 79.8 centimeters. And line segment πΆπΉ 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·πΈπΆπΉ as 302 centimeters.