The Grande Finale

Part Deux

Square Inscribed in a semicircle

I'm first going to start by construing the square of side length s inside a semicircle.

I know that if I construct a square around the circle this will help me to create another square.

Next I can find the inter section of the vertex of the square and the center of the circle, this intersection with the semi circle will be the length s from the diameter to in the intersection on the arc.

Thus we have what we were trying to create, and we also have a few extra bits we found along the way so that finding a and it's relation to s will be easier.

Click on the photo above so that you can se how a and s will change as the diameter changes in the circle.

Find

Starting from the photo above we can find the radius of the circle.

If we look at the pythagorean Theorem and see that

We know what the radius is from about so we can substitute for r.

Holy Cow, We have seen the golden ratio once again!

**We have not forgotten about the negative symbol, it has been omitted because a distance can not be negative.**