Answer:
x = 4[tex]\sqrt{6}[/tex] , y = 8[tex]\sqrt{2}[/tex]
Step-by-step explanation:
Using the sine ratio in the right triangle on the left and the exact value
sin45° = [tex]\frac{\sqrt{2} }{2}[/tex]
let the altitude of the outer triangle be h ( common to both right triangles )
sin45° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{h}{8}[/tex] = [tex]\frac{\sqrt{2} }{2}[/tex] ( cross- multiply )
2h = 8[tex]\sqrt{2}[/tex] ( divide both sides by 2 )
h = 4[tex]\sqrt{2}[/tex]
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Using the tangent ratio in the right triangle on the right and the exact value
tan30° = [tex]\frac{1}{\sqrt{3} }[/tex] , then
tan30° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{h}{x}[/tex] = [tex]\frac{4\sqrt{2} }{x}[/tex] = [tex]\frac{1}{\sqrt{3} }[/tex] ( cross- multiply )
x = 4[tex]\sqrt{6}[/tex]
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Using the sine ratio in the right triangle on the right and the exact value
sin30° = [tex]\frac{1}{2}[/tex] , then
sin30° = [tex]\frac{h}{y}[/tex] = [tex]\frac{4\sqrt{2} }{y}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
y =8[tex]\sqrt{2}[/tex]
