Second Diagram
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Givens
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Sin ADC = sin(180 - ADB)
Sin ADC = Sin(ADB) Sine Relationship
Call D the intersect of the angle bisector of A and the line BC. I'm assuming that is the angle bisector. If I'm wrong, leave me a note.
Sin ADC / Sin DAC = (2x - 5) / 14
Sin ADC = (2x - 5) * Sin(DAC) /14 . . . . .(1)
Sin ADB / Sin BAD = (x + 8) / 10
Sin ADB = (x + 8) * sin(BAD) / 10 . . . . . (2)
BAD = DAC Angle bisector.
Reduction
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(2x -5) * Sin(DAC) /14 = (x + 8) * Sin(DAB) / 10 The two sin functions are equal because they are the bisected angles of BAC.
(2x - 5) / 14 = (x + 8) / 10 Cross multiply
10*(2x - 5) = 14 * (x + 8)
20x - 50 = 14x + 112
6x = 162
x = 27
I have a feeling I've done this the hard way. An angle bisector probably divides the sides in a proportion like BA/10 = CA / 14. If that is true, you will get the same answer. I'll do the first one the same way I've suggested here.
Problem 1
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x/4 = 2.25 / 3
3x = 9
x = 3