SOLUTION: In triangle, ABC, angle ABC=90, and point D lie on segment BC such that AD is an angle bisector. If AB=105 and BD=42, then find AC.

Algebra ->  Angles -> SOLUTION: In triangle, ABC, angle ABC=90, and point D lie on segment BC such that AD is an angle bisector. If AB=105 and BD=42, then find AC.      Log On


   

Question 1117085: In triangle, ABC, angle ABC=90, and point D lie on segment BC such that AD is an angle bisector. If AB=105 and BD=42, then find AC.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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In triangle, ABC, angle ABC=90, and point D lie on segment BC such that AD is an angle bisector.
If AB=105 and BD=42, then find AC.
:
Draw out the triangle as described.
A smaller right triangle ABD is also formed.
A portion of the bisected angle (BAC) forms an angle (x) of this triangle.
Find this angle (x) using the tangent
tan(x) = 42%2F105
x = 21.8 degrees
the Angle BAC = 2 * 21.8 = 43.6 degrees
AC is the hypotenuse, find AC using the cosine of 43.6 degree
cos(43.8) = 105%2Fh
h = 105%2Fcos%2843.6%29
h = 145 = AC