Euclidean Circle Geometry. Calculating the Radius

Question

In the diagram below, DE is a chord of the circle that goes through c,d and e. A is the center of the circle. The perpendicular line from centre A intersects DE at B and the circle at C,

DE=100cm BC= 10cm AB=X

I need to calculate the length of AB / x.

Basically i need to calculate the radius of the circle without the diameter or circumference or anything. Do you have any suggestions as to how I would do this?

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Answer 1

You have $DB= 50$ and $AD=AC=x+10$

Now use Pythagoream theorem and find $x$

Answer 2

Extend $CA$ to cut the circle again at $F$. Then $BF=10+2x$ and by the intersecting chords property, $$10(10+2x)=50\times 50$$ Thus $x = 120$