### Video Transcript

Given that π΄π΅ is equal to 60,
π΄πΆ is 40, and π΅πΆ is 31, what is πΆπ·?

The diagram indicates that the two
angles shown are congruent. That is, theyβre the same number of
degrees or radians. The line segment π΄π· is then the
bisector of the exterior angle at π΄ of the triangle π΄π΅πΆ. We recall that the exterior angle
bisector theorem gives us the identity π·πΆ over π·π΅ is equal to π΄πΆ over π΄π΅,
that is, that the ratio of the length π·πΆ to the length π·π΅ is the same as the
ratio of the length π΄πΆ to the length π΄π΅. Now, we know that the length π·π΅
is equal to π·πΆ plus πΆπ΅. So we can replace this in our
formula. And we have π·πΆ over π·πΆ plus
πΆπ΅ is equal to π΄πΆ over π΄π΅.

Weβre given the length π΄π΅ is
equal to 60, π΅πΆ is equal to 31, and π΄πΆ is equal to 40. Into our equation then, the
equation has only one unknown; thatβs π·πΆ. And our equation is then π·πΆ
divided by π·πΆ plus 31 is 40 over 60. And notice since the length π·πΆ is
the same as πΆπ·, this is what weβre looking for. So now letβs solve our equation for
π·πΆ. Multiplying both sides by 60 and
also π·πΆ plus 31, we have 60π·πΆ is equal to 40 multiplied by π·πΆ plus 31. Distributing the parentheses on the
right, we have 40π·πΆ plus 1240. And now subtracting 40π·πΆ from
both sides, we have 20π·πΆ is 1240. Dividing both sides by 20, we have
π·πΆ is 62. And since π·πΆ is the same as πΆπ·,
we have πΆπ· is equal to 62 units.