### Video Transcript

In the triangle π΄π΅πΆ, π΄π΅ is 76
centimeters, π΄πΆ is 57 centimeters, and π΅π· is 52 centimeters. Given that the line segment π΄π·
bisects the angle π΄ and intersects the line segment π΅πΆ at π·, determine the
length of the line segment π΄π·.

We recall that if the line segment
π΄π· bisects the angle π΄ in a triangle, then we have the theorem π΄π· is equal to
the square root of π΄πΆ multiplied by π΄π΅ minus π·πΆ multiplied by π·π΅. Now from the question, we know that
π΄πΆ is equal to 57, π΅π·, which is π·π΅, is equal to 52, and that π΄π΅ has length
76. And so to find the length π΄π·, we
must first find the length π·πΆ. To do this, weβre going to use the
interior angle bisector theorem. This says that the bisector of an
interior angle of a triangle divides the opposite side into segments whose lengths
have the same ratio as the lengths of the noncommon adjacent sides of the bisected
angle.

What this means in our triangle is
that π·πΆ over π΅π· is the same as π΄πΆ over π΄π΅. Now we know that π΅π· is 52, π΄πΆ
is 57, and π΄π΅ is 76. So we have π·πΆ over 52 is equal to
57 over 76. Now, multiplying both sides by 52,
weβre left with π·πΆ on the left-hand side. And 57 divided by 76 multiplied by
52 is 39. So π·πΆ is 39 centimeters. We now have all the information we
need to calculate π΄π·. Substituting in our values, we have
π΄π· is the square root of 57 times 76 minus 39 times 52, that is, the square root
of 4332 minus 2028, which is the square root of 2304, that is, 48. The length of the line segment π΄π·
is therefore 48 centimeters.