In triangle 𝑋𝑌𝑍, the measure of angle 𝑌 is 90 degrees. 𝑌𝑍 equals 15. And 𝑋𝑍 equals 25. Triangle 𝐿𝑀𝑁 is similar to triangle 𝑋𝑌𝑍, where vertices 𝐿, 𝑀, and 𝑁 correspond to vertices 𝑋, 𝑌, and 𝑍, respectively. And each side of triangle 𝐿𝑀𝑁 is one-sixth the length of the corresponding side of triangle 𝑋𝑌𝑍. What is the value of cos 𝐿?
This question has quite a lot of information. So I think it’s helpful if we try and sketch what we know. We know that angle 𝑌 is 90 degrees. And that tells us we’re going to be working with a right triangle. Here’s our right triangle. Angle 𝑌 measures 90 degrees. And so we’ll label this triangle 𝑋𝑌𝑍. 𝑌𝑍 equals 15. And 𝑋𝑍 equals 25. We know that triangle 𝐿𝑀𝑁 is similar to triangle 𝑋𝑌𝑍. This means that triangle 𝐿𝑀𝑁 is also a right triangle. We know that 𝐿 corresponds to 𝑋. 𝑀 corresponds to 𝑌. And 𝑁 corresponds to 𝑍.
We are looking for cos of 𝐿. We know that cosine relationship equals the adjacent side over the hypotenuse. Something else we can note about similar triangles is that corresponding angles are congruent, which means that angle 𝑋 is equal to angle 𝐿. It also means the cos of 𝑋 equals the cos of 𝐿. Since we know two of the three sides of triangle 𝑋, 𝑌, and 𝑍, we can find the third side in triangle 𝑋, 𝑌, and 𝑍 to find the cos of 𝑋, which will be equal to the cos of 𝐿. Why would we do this? We know that, in triangle 𝐿𝑀𝑁, each side length is one-sixth the side length of triangle 𝑋𝑌𝑍. And that means the length of 𝐿𝑁 is twenty-five sixths, which we could reduce to four and one-sixth. Either way, dealing with this fraction to find three sides is not as nice as dealing with a whole number.
Now, we know that 𝑎 squared plus 𝑏 squared equals 𝑐 squared. We could set up this equation. 15 squared plus 𝑋𝑌 squared equals 25 squared. However, this is a special triangle. Notice that we have three times five. That equals 15. And we have five times five. That equals 25. We recognize that this triangle fits the three-four-five ratio. Three times five is 15. Five times five is 25. If we multiply four by our third side, we find that 𝑋𝑌 equals 20. Since we know the cos of 𝑋 equals the cos of 𝐿, we need the adjacent side to 𝑋, which measures 20, and the hypotenuse, which is 25. The cos of 𝑋 equals 20 over 25. And we can reduce this ratio by dividing both sides by five and say the cos of 𝑋 equals four over five.
Since these two triangles are similar, the cos of 𝐿 is also four out of five.