### Video Transcript

From the figure, determine the length of the line π΅π·. If necessary, round your answer to the nearest hundredth.

Weβre asked in this example to find the length of the line π΅π·, which we can see is the perpendicular projection to π· on the hypotenuse of the right triangle π΄π΅πΆ from the right angle at π΅. To find π΅π·, our first step will be to use the Pythagorean theorem to find the length of the hypotenuse π΄πΆ of triangle π΄π΅πΆ. We can then use this in one part of the right triangle altitude theorem to find the side length πΆπ· and then use this value in the Pythagorean theorem in right triangle πΆπ·π΅ to find the side length we want, which is π΅π·.

So, letβs start with the Pythagorean theorem applied to triangle π΄π΅πΆ to find the length of π΄πΆ. We have π΄πΆ squared equal to πΆπ΅ squared plus π΄π΅ squared. And substituting our two known side lengths, this gives 15 squared plus eight squared on the right-hand side. This evaluates to 289. And taking the positive square root on both sides, positive since lengths are positive, we have π΄πΆ equals the square root of 289, which is 17 centimeters.

So, now marking this on the diagram and clearing some space, next we can use the right triangle altitude theorem to find side length πΆπ·. We know that πΆπ΅ is 15 centimeters. And weβve just found that π΄πΆ is 17 centimeters. And so we have 15 squared equals πΆπ· multiplied by 17. Now, dividing through by 17 and evaluating 15 squared, we have πΆπ· equal to 225 over 17.

Next, applying the Pythagorean theorem to triangle πΆπ·π΅, we have π΅π· squared equals π΅πΆ squared minus πΆπ· squared. Thatβs 15 squared minus 225 over 17 all squared. This is 49.826 and so on. And taking the square root on both sides, we have π΅π· equal to 7.058 and so on. Finally, rounding to the nearest hundredth, which is to two decimal places, we have that π΅π· is equal to 7.06 centimeters.