### Video Transcript

Find the length of π΄πΉ approximating the result to the nearest hundredth.

In this example, weβre asked to find the length of π΄πΉ, where in right triangle π΄π΅π· π΄πΉ is the perpendicular projection from the right angle at π΄ to πΉ on side π΅π·.

The first thing we can do to get started is use the Pythagorean theorem to find the length of side π·π΅ in right triangle π΄π΅π·. In our case, this gives π΄π΅ squared plus π΄π· squared is equal to π·π΅ squared. That is, 33 squared plus 56 squared equals π·π΅ squared. Evaluating the left-hand side gives π·π΅ squared equal to 4225. And taking the positive square root on both sides, we have π·π΅ equal to 65.

Okay, so we have the side length π·π΅, but how does this help us to find π΄πΉ? Well, our next step is to use the right triangle altitude theorem to find the two side lengths π·πΉ and π΅πΉ. We can do this since we have lengths π΄π·, π΄π΅, and π·π΅, which is the same as π΅π·. And thatβs all of the other lengths involved in these two equations. And the reason we want to do this is so we can use π·πΉ and π΅πΉ in the corollary of the altitude theorem to find the length we want, which is π΄πΉ.

So letβs start with the first equation. Thatβs π΄π· squared equals π·πΉ times π·π΅. In our case, this gives us 56 squared equals π·πΉ times 65. Then, evaluating the square on the left-hand side and dividing both sides by 65, we find π·πΉ equal to 48.2461 and so on. Next, looking at the second equation in the altitude theorem, π΄π΅ squared equals π΅πΉ multiplied by π΅π·, we have 33 squared equal to π΅πΉ times 65. Again, evaluating the square on the left and dividing through by 65, we have that π΅πΉ is equal to 16.7538 and so on.

So now we have π·πΉ and π΅πΉ, which are the two values we need to find the length π΄πΉ using the corollary. Substituting these values in, we have π΄πΉ squared equal to 48.2461 multiplied by 16.7538. This is 808.3086 and so on. And now taking the positive square root on both sides, we have π΄πΉ equal to 28.4307.

And so since weβre asked to approximate the length of π΄πΉ to the nearest hundredth, which is to two decimal places, using the Pythagorean theorem together with the right triangle altitude theorem and its corollary, we find the length π΄πΉ to the nearest hundredth equals 28.43 centimeters.