Question Video: Finding the Length of the Hypotenuse Using the Properties of the Medians of Right-Angled Triangles Mathematics

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.