# Geometry; Ball in a triangle

• Nov 14th 2012, 01:22 PM
doktorQ
Geometry; Ball in a triangle
Hey, I need some help on this problem.(check attachment)

The question goes something like this:
"What is the size of the yellow area".

Can anyone help me with it?
Also if you actually know what to do, could you show me it in a kind of step-by-step as I'm not so good at all the technical terms(translating them)
• Nov 14th 2012, 01:51 PM
Re: Geometry; Ball in a triangle

I labeled key vertices for convenience. Since OA is a radius meeting a point of tangency angle OAB is a right angle. Hence we can use the Pythagorean Theorem to find X. (OB is the hypotenuse) Since we have all three side lengths, we can calculate the area of triangle OAB directly.

Next we can find the area of the light blue wedge AOC. We can find the angle AOC using triangle OAB as reference using trigonometry. Then use the area formula that involves the angle and the radius OA to find the area of the light blue wedge

Triangle OAB - Wedge AOC = half the area of the desired region. Hence
2 (Triangle OAB - Wedge AOC) = the area of the desired region.
• Nov 14th 2012, 02:14 PM
doktorQ
Re: Geometry; Ball in a triangle
I see you point, you probably though there is two 2x5 triangles in the drawing. Which can be solver by pythagoras, but thats wrong(I had the same initial idea too)

But as you may see if you think about it, there is no right angle triangle, the triangle would have to be infinatley long for that to work.
The line from the triangle do not cut the circle at the top or bottom, but slighly before that.
• Nov 14th 2012, 02:34 PM
Re: Geometry; Ball in a triangle
In other words, are you saying that AB and the other line symmetric about OB are not tangent lines?
• Nov 14th 2012, 03:05 PM
Plato
Re: Geometry; Ball in a triangle
Quote:

Originally Posted by doktorQ
Hey, I need some help on this problem.(check attachment)
The question goes something like this:
"What is the size of the yellow area".

Using the re-labeled diagram in reply #2, we can find the area of the circular sector bounded by $\displaystyle A,~O,~C$.

It is $\displaystyle 0.5R^2\theta$ where $\displaystyle R=2~\&~\theta=\angle AOC$
But $\displaystyle \theta=\arccos \left( {\frac{2}{5}} \right)$.

You can find the area of $\displaystyle \Delta AOB$