An Angle Inscribed in a Semicircle

A semicircle is half a circle.  If you choose a point P on the semicircle and draw segments to the endpoints of the diameter, you end up with an angle inscribed in a semicircle:

 

Now it looks like a right angle.  Let's prove it is a right angle in a special case.  To prove it in general requires a strong foundation in algebra, so we won't attempt that here.

Let's suppose the circle has radius 10.  We place the semicircle at the origin, and suppose the x-coordinate of P is 6.  We will call the y-coordinate b for the time being.

We need to know the y-coordinate of P.  The equation of a circle of radius r centered at the origin is x² + y² = r², and since P is on the circle, the coordinates 6 and b are related by this equation, 6² + b² = 10².  Solving for b we get b = 8.  So P has coordinates (6, 8).

 

Now to prove the angle at P is a right angle, we can show the slopes of lines AP and BP are opposite reciprocals:

So the slopes are opposite reciprocals, and therefore the angle at P is a right angle

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