Answer:
see explanation
Step-by-step explanation:
Using the exact trigonometric values
cos30° = [tex]\frac{\sqrt{3} }{2}[/tex], sin30° = [tex]\frac{1}{2}[/tex] and tan45° = 1
Let the point where the segment from C meets AB be D, then
Using ΔBCD
cos30° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{BD}{x}[/tex] and
[tex]\frac{BD}{x}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )
2BD = [tex]\sqrt{3}[/tex] x ( divide both sides by 2 )
BD = [tex]\frac{\sqrt{3} }{2}[/tex] x ←
sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{CD}{x}[/tex] and
[tex]\frac{CD}{x}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2CD = x ( divide both sides by 2 )
CD = [tex]\frac{1}{2}[/tex] x
Using ΔACD
tan45° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{CD}{AD}[/tex] = [tex]\frac{\frac{x}{2} }{AD}[/tex] = 1 , hence
AD = [tex]\frac{1}{2}[/tex] x ←
Hence
AB = BD + AD = [tex]\frac{\sqrt{3} }{2}[/tex] x + [tex]\frac{1}{2}[/tex] x = ([tex]\frac{\sqrt{3}+1 }{2}[/tex] ) x
