# Question Video: Finding an Unknown Side Length in a Triangle Using the Angle Bisector Theorem Mathematics

In the triangle π΄π΅πΆ, π΄π΅ = 76 cm, π΄πΆ = 57 cm, and π΅π· = 52 cm. Given that the line segment π΄π· bisects the angle π΄ and intersects the line segment π΅πΆ at π·, determine the length of the line segment π΄π·.

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### Video Transcript

In the triangle π΄π΅πΆ, π΄π΅ is 76 centimeters, π΄πΆ 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΅πΆ 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΄πΆ multiplied by π΄π΅ minus π·πΆ multiplied by π·π΅. Now from the question, we know that π΄πΆ 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·πΆ. To do this, weβre 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·πΆ over π΅π· is the same as π΄πΆ over π΄π΅. Now we know that π΅π· is 52, π΄πΆ is 57, and π΄π΅ is 76. So we have π·πΆ over 52 is equal to 57 over 76. Now, multiplying both sides by 52, weβre left with π·πΆ on the left-hand side. And 57 divided by 76 multiplied by 52 is 39. So π·πΆ 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.