### Video Transcript

In the figure below, the measure of
angle πππ is equal to the measure of angle πππ, which is equal to 90
degrees. π is the midpoint of line segment
ππ, and the measure of angle π is 30 degrees. Given that ππ equals 13
centimeters, find the length of line segment ππ.

Letβs start by taking some
information from the problem and labeling our figure. We know that πππ is 90
degrees. We also know that πππ is 90
degrees. π is the midpoint of line segment
ππ. And ππ is the hypotenuse of
triangle πππ. We have the measure of angle π is
30 degrees. Thatβs already labeled here. And then, we know that ππ is 13
centimeters. But because we have a midpoint, we
can say that each of these midsegments is six and a half centimeters.

Because π is a midpoint of line
segment ππ and line segment ππ falls on this right triangle as the hypotenuse,
we can say that the line segment ππ is a median of this right triangle. Which should remind us of the
property that in a right triangle, the length of the median from the vertex of the
right angle is equal to half the length of the hypotenuse. The hypotenuse was 13. Half that length is six and a half,
which means that the line segment ππ is equal to 6.5 centimeters.

Line segment ππ is not the side
weβre trying to find. We want to know the length of
ππ. So now, we need to focus on what we
know about triangle πππ. We know that one of the angles is
30 degrees. One is 90, making the other 60. That means triangle πππ is a
30-60-90 triangle. And we should remember that for any
30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse. Or put another way, the ratio of
side lengths for a 30-60-90-degree triangle occurs in one to square root of three to
two, where the smallest side is opposite the 30-degree angle and the largest is the
hypotenuse. The hypotenuse will be two times
the side length of the smallest side in a 30-60-90 triangle.

ππ is the hypotenuse of triangle
πππ. So, we can say that the length of
ππ will be two times the length of ππ because ππ is the side opposite the
30-degree angle and is, therefore, the smallest side length in this 30-60-90-degree
triangle. Thatβs two times 6.5, which is 13
centimeters. The hypotenuse of triangle πππ
measures 13 centimeters.