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In triangle ABC, we have angle BAC = 60 degrees and angle ABC = 45 degrees. The bisector of angle A intersects line BC at point T, and AT = 24. What is the length of side BC?

 

 Dec 17, 2021
 #1
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BC = 12 + 25*sqrt(3).

 Dec 17, 2021
 #2
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This is how to solve ABC:

In my estimations you have a triangle with 2 angles and an altitude or height.  You have to find the base of the triangle in order to find the area.  I set up the triangle and sine the 60 degree angle was bisected I know that the vertex angle is 30 in my right triangle.  I used the tangent ratio of 30 to find the base of that triangle and then multiplied it by 2 to get that the whole base measure is 27.71281292.  Then I multiplied it by the height I was given of 24 and divided the whole mess in half to get the area.  332.553 units squared.

 

Somewhere in my explanation I gave a hint to the legnth of BC.

 

🎅🎅🎅🎅🎅

🥳🥳🥳🥳🥳

 Dec 17, 2021
 #3
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BC = 29.39387691  (!!!)

 

Triangle ACT is an isosceles triangle and side AC = 24.

 

CT = 2(24 * sin15)

 

By using the law of sines, we can calculate side BT.

 Dec 17, 2021

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