Find the length of line segment 𝑌𝑍.
Line segment 𝑌𝑍 is here, and we can mark it with a lowercase 𝑥. We see that we’re dealing with two right triangles, both of which have one 30-degree angle, which reminds us that they are 30-60-90 triangles. We’ll consider a few different methods for finding this missing side length.
The first way is to remember the 30-60-90 side length ratios. We sometimes call this 30-60-90 triangle a special right triangle. The side lengths occur in the ratio one to square root of three to two. We know the longest side is the hypotenuse. The smallest side will be the side opposite the 30-degree angle. And the other side is the side length opposite the 60-degree angle.
In order to use the ratios, you have to know at least one of the side lengths. In our smaller triangle, we currently know none of the side lengths. But in our larger triangle, we know the length of the hypotenuse, and we also know that the large triangle and the small triangle share the side length 𝑋𝑍. If we use the ratios to find the side length 𝑋𝑍, that will help us find our missing side length in the smaller triangle. For our large triangle, the hypotenuse is 16.7 centimeters, and the side length we want to find is opposite the 30-degree angle. Let’s call it 𝑦.
If we’re considering the ratio of the hypotenuse to the smallest side, it would be two to one. To go from the hypotenuse’s side length to the smallest side length, you multiply by one-half. And that means the side length we’ve labeled 𝑦 will be one-half of 16.7. It will be 8.35. Now, we have one of the side lengths for our smaller triangle. In our smaller triangle, the hypotenuse is 8.35. And the missing side length will be the side length opposite the 30-degree angle. And that means this 𝑥-value, the side length of 𝑌𝑍, will be one-half 8.35, which will be 4.175. And all the side lengths are being measured in centimeters.
But I did say that we would consider a few different methods. This method only works if you remember the ratios of side lengths in a 30-60-90 triangle. If you didn’t remember those ratios, you could use right-triangle trigonometry. Starting with our large triangle, we know the hypotenuse. And we want to find the side length opposite the 30-degree angle. We want to do this so that we know one of the sides of the smaller triangle. We know that the sine relationship is the opposite over the hypotenuse, which means we could say sin of 30 degrees equals this missing 𝑦-value over the hypotenuse 16.7.
To solve for 𝑦, we would multiply 16.7 by both sides of the equation. 16.7 times sin of 30 degrees equals 8.35. Now that we know that the 𝑋𝑍 length is 8.35 centimeters, we sketch the smaller triangle. The value we’re looking for is opposite the 30-degree angle, which means we can use the sine relationship a second time. In this case, the opposite side will be 𝑥 and the hypotenuse of the smaller triangle is 8.35. From there, we multiply both sides of the equation by 8.35. 8.35 times sin of 30 degrees is 4.175. The second trigonometry method confirms what we found at first, that the length of 𝑌𝑍 is 4.175 centimeters.