### Determining Hypotenuse Length

Learn how to determine hypotenuse length in right triangles with this step-by-step explanation and illustration of the Pythagorean theorem.

Practice exercises and free samples for the math test are provided on this page.

**Problem 1:**

If one leg of a triangle is 5cm and the other leg is 12cm, what is the measurement of the hypotenuse of the triangle?

A. 5√12cm

B. 12√5cm

C. √17cm

D. 13cm

E. 17cm

**The correct answer is D.**

For any right triangle with sides A, B, and C, you need to remember this formula:

√A^{2} + B^{2} = C

In other words, the length of the hypotenuse (represented by side C) is equal to the square root of the sum of the squares of the other two sides of the triangle.

So, we can substitute the values into the formula in order to find the solution for this problem:

√A^{2} + B^{2} = C

√5^{2} + 12^{2} = C

√25 + 144

√169

13cm

**Problem 2:**

In the figure below, ∠Y is a right angle and ∠X = 60°.

If line segment YZ is 5 units long, then how long is line segment XY?

A. 5 units

B. 6 units

C. 15 units

D. ^{5}/_{√3} units

E. 30 units

**The correct answer is D.**

Triangle XYZ is a 30° – 60° – 90° triangle.

Using the Pythagorean theorem, its sides are therefore in the ratio of 1: √3: 2

In other words, using relative measurements, the line segment opposite the 30° angle is 1 unit long, the line segment opposite the 60° angle is √3 units long, and the line segment opposite the right angle (the hypotenuse) is 2 units long.

In this problem, line segment XY is opposite the 30° angle, so it is 1 proportional unit long.

Line segment YZ is opposite the 60° angle, so it is √3 proportional units long.

Line segment XZ (the hypotenuse) if the angle opposite the right angle, so it is 2 proportional units long.

So, in order to keep the measurements in proportion, we need to proceed as follows:

XY ÷ YZ = ^{1}/_{√3}

Now substitute the known measurement of YZ from the above figure, which is 5 in this problem.

XY ÷ 5 = ^{1}/_{√3}

XY ÷ 5 × 5 = ^{1}/_{√3} × 5

XY = ^{5}/_{√3}