Question Video: Finding Unknown Side Lengths in a Triangle Using the Angle Bisector Theorem | Nagwa Question Video: Finding Unknown Side Lengths in a Triangle Using the Angle Bisector Theorem | Nagwa

# Question Video: Finding Unknown Side Lengths in a Triangle Using the Angle Bisector Theorem Mathematics • First Year of Secondary School

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Given that π΄π΅ = 60, π΄πΆ = 40, and π΅πΆ = 31, what is πΆπ·?

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### Video Transcript

Given that π΄π΅ is equal to 60, π΄πΆ is 40, and π΅πΆ is 31, what is πΆπ·?

The diagram indicates that the two angles shown are congruent. That is, theyβre 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΅πΆ. We recall that the exterior angle bisector theorem gives us the identity π·πΆ over π·π΅ is equal to π΄πΆ over π΄π΅, that is, that the ratio of the length π·πΆ to the length π·π΅ is the same as the ratio of the length π΄πΆ to the length π΄π΅. Now, we know that the length π·π΅ is equal to π·πΆ plus πΆπ΅. So we can replace this in our formula. And we have π·πΆ over π·πΆ plus πΆπ΅ is equal to π΄πΆ over π΄π΅.

Weβre given the length π΄π΅ is equal to 60, π΅πΆ is equal to 31, and π΄πΆ is equal to 40. Into our equation then, the equation has only one unknown; thatβs π·πΆ. And our equation is then π·πΆ divided by π·πΆ plus 31 is 40 over 60. And notice since the length π·πΆ is the same as πΆπ·, this is what weβre looking for. So now letβs solve our equation for π·πΆ. Multiplying both sides by 60 and also π·πΆ plus 31, we have 60π·πΆ is equal to 40 multiplied by π·πΆ plus 31. Distributing the parentheses on the right, we have 40π·πΆ plus 1240. And now subtracting 40π·πΆ from both sides, we have 20π·πΆ is 1240. Dividing both sides by 20, we have π·πΆ is 62. And since π·πΆ is the same as πΆπ·, we have πΆπ· is equal to 62 units.

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