# Video: US-SAT03S3-Q20-505140892795

In triangle πππ, the measure of β π is 90Β°. ππ = 15 and ππ = 25. Triangle πΏππ is similar to triangle πππ, where vertices πΏ, π, and π correspond to vertices π, π, and π, respectively, and each side of triangle πΏππ is one-sixth the length of the corresponding side of triangle πππ. What is the value of cos πΏ?

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### Video Transcript

In triangle πππ, the measure of angle π is 90 degrees. ππ equals 15. And ππ equals 25. Triangle πΏππ is similar to triangle πππ, where vertices πΏ, π, and π correspond to vertices π, π, and π, respectively. And each side of triangle πΏππ is one-sixth the length of the corresponding side of triangle πππ. What is the value of cos πΏ?

This question has quite a lot of information. So I think itβs helpful if we try and sketch what we know. We know that angle π is 90 degrees. And that tells us weβre going to be working with a right triangle. Hereβs our right triangle. Angle π measures 90 degrees. And so weβll label this triangle πππ. ππ equals 15. And ππ equals 25. We know that triangle πΏππ is similar to triangle πππ. This means that triangle πΏππ is also a right triangle. We know that πΏ corresponds to π. π corresponds to π. And π corresponds to π.

We are looking for cos of πΏ. 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 π is equal to angle πΏ. It also means the cos of π equals the cos of πΏ. Since we know two of the three sides of triangle π, π, and π, we can find the third side in triangle π, π, and π to find the cos of π, which will be equal to the cos of πΏ. Why would we do this? We know that, in triangle πΏππ, each side length is one-sixth the side length of triangle πππ. And that means the length of πΏπ 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 π squared plus π squared equals π squared. We could set up this equation. 15 squared plus ππ 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 ππ equals 20. Since we know the cos of π equals the cos of πΏ, we need the adjacent side to π, which measures 20, and the hypotenuse, which is 25. The cos of π equals 20 over 25. And we can reduce this ratio by dividing both sides by five and say the cos of π equals four over five.

Since these two triangles are similar, the cos of πΏ is also four out of five.