### Video Transcript

Find the length of π΅π·.

So, weβre trying to find this length and we can do so using similar right triangles. Two right triangles are similar to each other if an acute angle in one triangle is equal to the measure of an acute angle in the other triangle. Triangle π΄π΅πΆ and triangle π·π΅π΄ each have a right angle at angle π΅π΄πΆ and at angle π΅π·π΄. And angle π΅π·π΄ is equal to 90 degrees because angle π΄π·πΆ β this one β is 90 degrees. And since theyβre right next to each other and make a straight line, angle π΅π·π΄ is 90 degrees.

So and these triangles both share a common angle measure at angle π΅. They share that angle. Therefore, triangle π΄π΅πΆ and triangle π·π΅π΄ are similar and their side lengths are proportional to each other. So, we can set up a proportion: π΅π· is to π΅π΄ as π΄π΅ is to πΆπ΅.

And we can solve proportions by cross multiplying and solving. So, we have π΅π· times πΆπ΅ is equal to π΅π΄ times π΄π΅. But those are the exact same side. So, we can write it as π΄π΅ squared.

So, letβs begin plugging in some side lengths. We know that π΄π΅ is equal to 15 centimeters. We donβt know πΆπ΅. And π΅π· is actually what weβre trying to solve for. So, is there a way that we can find the length of πΆπ΅? We can, by using the Pythagorean theorem because triangle π΄π΅πΆ is a right triangle.

The Pythagorean theorem states the square of the longest side is equal to the sum of the squares of the shorter sides. So, out of triangle π΄π΅πΆ, the longest side is across from the 90-degree angle, which would be πΆπ΅, making the two shorter sides 15 and 20. 15 squared is 225 and 20 squared is 400. Adding them together, we get 625.

And to solve for πΆπ΅, we need to square root both sides. Therefore, πΆπ΅ is equal to 25. And now we can plug this in. So, letβs go ahead and square 15. And we get 225.

Now to solve for π΅π·, we need to divide both sides of the equation by 25. The 25s cancel on the left. And we find that π΅π· is equal to nine. Therefore, the length of π΅π· is equal to nine centimeters.