### Video Transcript

Find the length of the side π΄π΅.

To find the length of side π΄π΅ in triangle π΄π΅πΆ, we note first that this is a right triangle at π΄ and that π· is the perpendicular projection of π΄ onto side π΅πΆ. This being the case, we recall the right triangle altitude theorem, which we can use to find the length of side π΄π΅. This gives us a formula for each of the squares of sides π΅π΄ and πΆπ΄. And we see itβs the first of these that we can use to find side length π΄π΅, since this is the same as π΅π΄.

We have π΅π΄ squared equals π΅π· multiplied by π΅πΆ, where we see that π΅πΆ is equal to the sum of π΅π· and πΆπ·. Weβre given these lengths in the diagram. π΅π· equals 9.6 and πΆπ· equals 5.4 centimeters. So, we have π΅π΄ squared equal to 9.6 multiplied by the sum of 9.6 and 5.4. Thatβs 9.6 times 15, which is equal to 144. So, we have π΅π΄ squared equal to 144. And taking the positive square root on both sides, positive since lengths are positive, we have π΅π΄ equal to 12.

Since π΅π΄ is the same as π΄π΅, using the right triangle altitude theorem, weβve been able to find the length of side π΄π΅, which is equal to 12 centimeters.