Answer:
a = [tex]6*\sqrt{3}[/tex]
b = 12
c = [tex]6\sqrt{2}[/tex]
Step-by-step explanation:
Since the triangles are right triangles with 60 and 45 degree angles, their side lengths follow special triangles.
A 45-45-90 right triangle has side lengths [tex]1-1-\sqrt{2}[/tex].
A 30-60-90 right triangle has side lengths [tex]1 - \sqrt{3} -2[/tex].
Starting with the top triangle which has a 60 degree angle, its side length 6 corresponds to a side length of 1 in the special triangle. It is 6 times bigger so its remaining sides will be 6 times bigger too.
Side a corresponds to side length [tex]\sqrt{3}[/tex]. Therefore, [tex]a = 6*\sqrt{3}[/tex].
Side b corresponds to side length 2, b = 2*6 = 12.
The bottom triangle has a 45 degree angle, its side length b= 12 corresponds to [tex]\sqrt{2}[/tex]. This means [tex]\sqrt{2}[/tex] was multiplied by [tex]12 = \sqrt{2} * \sqrt{72}[/tex]. This means that side c is [tex]\sqrt{72}=6\sqrt{2}[/tex].
