Video Transcript
In the figure below, the measure of
angle 𝑋𝑌𝑍 is equal to the measure of angle 𝑀𝑌𝑁, which is equal to 90
degrees. 𝑀 is the midpoint of line segment
𝑋𝑍, and the measure of angle 𝑁 is 30 degrees. Given that 𝑋𝑍 equals 13
centimeters, find the length of line segment 𝑀𝑁.
Let’s start by taking some
information from the problem and labeling our figure. We know that 𝑋𝑌𝑍 is 90
degrees. We also know that 𝑀𝑌𝑁 is 90
degrees. 𝑀 is the midpoint of line segment
𝑋𝑍. And 𝑋𝑍 is the hypotenuse of
triangle 𝑋𝑌𝑍. We have the measure of angle 𝑁 is
30 degrees. That’s already labeled here. And then, we know that 𝑋𝑍 is 13
centimeters. But because we have a midpoint, we
can say that each of these midsegments is six and a half centimeters.
Because 𝑀 is a midpoint of line
segment 𝑋𝑍 and line segment 𝑋𝑍 falls on this right triangle as the hypotenuse,
we can say that the line segment 𝑌𝑀 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 𝑌𝑀 is equal to 6.5 centimeters.
Line segment 𝑌𝑀 is not the side
we’re trying to find. We want to know the length of
𝑀𝑁. So now, we need to focus on what we
know about triangle 𝑀𝑌𝑁. We know that one of the angles is
30 degrees. One is 90, making the other 60. That means triangle 𝑀𝑌𝑁 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.
𝑀𝑁 is the hypotenuse of triangle
𝑀𝑌𝑁. So, we can say that the length of
𝑀𝑁 will be two times the length of 𝑀𝑌 because 𝑀𝑌 is the side opposite the
30-degree angle and is, therefore, the smallest side length in this 30-60-90-degree
triangle. That’s two times 6.5, which is 13
centimeters. The hypotenuse of triangle 𝑀𝑌𝑁
measures 13 centimeters.
