Hard geometry problem doubts

GRE Quant Problem Solving
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How did he know that F is directly upwards from C? (how did he know it would make a 90 deg angle?)

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If the diameter is 8, then CD and CF must each be 4.

I don’t think you can have a triangle with the sides 4, 4, 4\sqrt{2} in that ration of x, x, x\sqrt{2} without it being a right-angle triangle (specifically a 90-45-45) one. And the angle opposite the hypotenuse must be 90 degrees.

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