# A question about simple circles and triangles

Today I've encountered a question like the following I am adding a picture because I have to;

The question paragraph says;

$\text{Given} \quad |OF|=6 \quad \text{and} \quad |BF|=4$

What is $|CH|=x$

My Attempts;

I have noticed that the diameter $r=10$ (1)

I have drawn a line from $C$ to $O$ which also is $r$ (2)

I have written $|HO|=\sqrt{100-x^2}$ but couldn't go further,

What do you suggest?

• How about $\angle EOF$? Sep 19, 2017 at 15:30
• Do you imply that it is $90^\circ$? It didn't work $8$ is the wrong answer:( Sep 19, 2017 at 15:32
• Do you mean the diameter or the radius is 10? Sep 19, 2017 at 15:39
• $|OB|=10$ so $|OC|=10$ too, was the diameter $2r$ or $r$ I probably remembered it wrongly. Sep 19, 2017 at 15:40
• Something more is needed here. Is it also given that EC=ED=EF, as marked in the figure? Sep 19, 2017 at 16:06

Let $\alpha=\angle HFC$ and $a=EF$, so that $x=2a\sin\alpha$.

From the cosine law applied to triangle $OFC$ we get: $$OF^2+FC^2-2OF\cdot FC\cos\alpha=OC^2, \quad\hbox{that is:}\quad a^2-6a\cos\alpha=16.$$

From the cosine law applied to triangle $OFD$ (notice that $FD=\sqrt2a$) we get: $$OF^2+FD^2-2OF\cdot FD\cos(\pi/4+\alpha)=OD^2, \quad\hbox{that is:}\quad a^2-6a\cos\alpha+6a\sin\alpha=32.$$ Substituting here our previous result we thus get $6a\sin\alpha=16$, that is $x=2a\sin\alpha={16\over3}$.

• Is it ODF instead of EFD and I got lost in The part where you said $\cos(\pi/4+\alpha)$ why added $\alpha$. Excuse me for my ignorance and many thanks for this elegant solution:)) Sep 19, 2017 at 16:38
• @Deniz: $\angle DFE=\pi/4$ from triangle $EFD$ Sep 19, 2017 at 16:48
• $\angle OFD=\angle OFE+\angle EFD=\alpha+\pi/4$. And then you need of course the cosine addition formula. Sep 19, 2017 at 16:56
• I know the cosine sum formula, but I carelessly thought that $\pi/4=90^\circ$ I then realised it was $45^\circ$, thank you:) Sep 19, 2017 at 17:04

This is only a partial solution, because I get the results using analytic geometry

Anyway I hope it can be useful

• Which software do you use to draw these figures? Sep 19, 2017 at 17:01
• @MathLover GeoGebra at geogebra.org It is much much more than a tool for drawing! Sep 19, 2017 at 17:08
• Thanks @Raffaele. Sep 19, 2017 at 17:10