MHB Find the radius of the small circle O_2

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The radius of the small circle O_2 is determined to be 6 units. This conclusion is reached by applying the cosine rule to a triangle formed by the circle's center and the endpoints of a baseline. The sides of the triangle are defined in relation to the radius of the green circle. The calculations involve the cosine of an angle, leading to the simplified equation that confirms the radius. The discussion also includes light-hearted commentary about avatars and imaginary concepts, but the primary focus remains on the mathematical solution.
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find the radius of the small circle O_2:
View attachment 1660
 

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Re: find the radius of the small circle O_2

We can draw a triangle from the left bottom corner, to the center of the small circle, to the right bottom corner.
And then split it into 2 rectangular triangles.

Let's call the radius of the small circle x.
Then the left rectangular triangle has hypotenuse (36-x) and horizontal side at the x-axis (x).
And the right rectangular triangle has hypotenuse (36+x) and horizontal side at the x-axis (36-x).

Since they share their third side, the following equation must hold (Pythagoras):
$$(36-x)^2 - x^2 = (36+x)^2 - (36-x)^2$$
$$(36+x)^2 - 2(36-x)^2 + x^2 = 0$$
$$(36^2+2\cdot 36 x +x^2) - 2(36^2-2\cdot 36 x + x^2) + x^2 = 0$$
$$6\cdot 36 x = 36^2$$
$$x = 6$$
$\blacksquare$
 
Re: find the radius of the small circle O_2

Albert said:
find the radius of the small circle O_2:
https://www.physicsforums.com/attachments/1660
[sp]
Let $r$ be the radius of the green circle. Draw the triangle whose vertices are the two ends of the red baseline and the centre of the green circle (the points labelled O, O_2 and O-1 in the diagram). The lengths of its sides are $36-r$, $36+r$ and $36$. The angle labelled $\theta$ has $\cos\theta = \dfrac r{36-r}$. The cosine rule then gives the equation $(36+r)^2 = 36^2 + (36-r)^2 -2\cdot36(36-r)\dfrac r{36-r},$ which simplifies to $r=6.$[/sp]

Edit. Yet again, ILS got there first (and I never even noticed).
 

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Re: find the radius of the small circle O_2

Opalg said:
Edit. Yet again, ILS got there first (and I never even noticed).

Must be my new avatar.
Sometimes it makes me feel imaginary.
 
Re: find the radius of the small circle O_2

I like Serena said:
Must be my new avatar.
Sometimes it makes me feel imaginary.

The invisible : not recognizable neither by name nor by avatar living in his imaginary complex paradigm , just kidding (Punch) .
 
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