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

In the figure, π‘šβˆ π‘‹π‘Œπ‘ = π‘šβˆ π‘€π‘Œπ‘ = 90Β°, 𝑀 is the midpoint of line segment 𝑋𝑍, and π‘šβˆ π‘ = 30Β°. Given that 𝑋𝑍 = 13 cm, find the length of line segment 𝑀𝑁.

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Video Transcript

In the figure below, the measure of angle π‘‹π‘Œπ‘ is equal to the measure of angle π‘€π‘Œπ‘, 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 π‘‹π‘Œπ‘ is 90 degrees. We also know that π‘€π‘Œπ‘ is 90 degrees. 𝑀 is the midpoint of line segment 𝑋𝑍. And 𝑋𝑍 is the hypotenuse of triangle π‘‹π‘Œπ‘. 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 π‘Œπ‘€ 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 π‘Œπ‘€ is equal to 6.5 centimeters.

Line segment π‘Œπ‘€ 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 π‘€π‘Œπ‘. We know that one of the angles is 30 degrees. One is 90, making the other 60. That means triangle π‘€π‘Œπ‘ 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 π‘€π‘Œπ‘. So, we can say that the length of 𝑀𝑁 will be two times the length of π‘€π‘Œ because π‘€π‘Œ 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 π‘€π‘Œπ‘ measures 13 centimeters.

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