# Euclidean Circle Geometry. Calculating the Radius

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?

enter image description here

• DB = half DE. radius $r = 10+x$. Solve $r^2=x^2+DB^2$. – Mauro ALLEGRANZA Sep 15 '16 at 8:42

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 $x = 120$

• Very good method. +1 – N.S.JOHN Sep 15 '16 at 8:47
• @N.S.JOHN Thanks. – user348749 Sep 15 '16 at 8:57

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

Now use Pythagoream theorem and find $x$

Let $r$ be the radius of the circle, and $x$ be AB.

From the diagram, $x^2 + 50^2 = r^2$, since every chord perpendicular to a diameter is also bisected by it. Furthermore, $x+10=r$.

Substituting, we have that $x^2 + 50^2 = (x+10)^2 = x^2 + 100 + 20x$. Hence, on simplifying we see that $x =120$cm. Hence, $r =130$cm.