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In the lower triangle, one of the angles of which is 45 degrees, we know:
In the triangle, the side facing the angle 45° is √2 / 2 times the hypothenuse.
Thus ;
[tex]10 = hypothenuse \times \frac{ \sqrt{2} }{2} \\ [/tex]
[tex]hypothenuse \times \frac{ \sqrt{2} }{2} = 10 \\ [/tex]
Multiply sides by 2
[tex]hypothenuse \times \sqrt{2} = 2 \times 10[/tex]
Divide sides by √2
[tex]hypothenuse = \frac{2 \times 10}{ \sqrt{2} } \\ [/tex]
[tex]hypothenuse = \frac{ \sqrt{2} \times \sqrt{2} \times 10 }{ \sqrt{2} } \\ [/tex]
[tex]hypothenuse = 10 \sqrt{2} [/tex]
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The common side of the triangles is hypothenuse for both.
[tex]hypothenuse \times \cos(60) = x[/tex]
[tex]10 \sqrt{2} \times \frac{1}{2} = x \\ [/tex]
[tex]x = \frac{2 \times 5 \sqrt{2} }{2} \\ [/tex]
[tex]x = 5 \sqrt{2} [/tex]
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The correct answer is the Third Option .
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