- #1
Albert1
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find the radius of the small circle O_2:
View attachment 1660
View attachment 1660
In summary, we use the cosine rule to find the radius of the small circle O_2 in the given diagram, which is $6$. This is in response to a conversation about finding the radius of the small circle O_2. One user jokingly mentions their invisible avatar causing them to feel imaginary.
- #2
I like Serena
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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):
$$(36-x)^2 - x^2 = (36+x)^2 - (36-x)^2$$
$$(36+x)^2 - 2(36-x)^2 + x^2 = 0$$
$$(36^2+2\cdot 36 x +x^2) - 2(36^2-2\cdot 36 x + x^2) + x^2 = 0$$
$$6\cdot 36 x = 36^2$$
$$x = 6$$
$\blacksquare$
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):
$$(36-x)^2 - x^2 = (36+x)^2 - (36-x)^2$$
$$(36+x)^2 - 2(36-x)^2 + x^2 = 0$$
$$(36^2+2\cdot 36 x +x^2) - 2(36^2-2\cdot 36 x + x^2) + x^2 = 0$$
$$6\cdot 36 x = 36^2$$
$$x = 6$$
$\blacksquare$
- #3
Opalg
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Re: find the radius of the small circle O_2
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).
[sp]Albert said:
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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- #4
I like Serena
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Re: find the radius of the small circle O_2
Must be my new avatar.
Sometimes it makes me feel imaginary.
Opalg said:
Must be my new avatar.
Sometimes it makes me feel imaginary.
- #5
alyafey22
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Re: find the radius of the small circle O_2
The invisible : not recognizable neither by name nor by avatar living in his imaginary complex paradigm , just kidding (Punch) .
I like Serena said:
The invisible : not recognizable neither by name nor by avatar living in his imaginary complex paradigm , just kidding (Punch) .
FAQ: Find the radius of the small circle O_2
How do you find the radius of a small circle?
To find the radius of a small circle, you need to measure the distance from the center of the circle to any point on the circle's edge. This distance is the radius, and it can be measured using a ruler or a compass.
What is the formula for finding the radius of a small circle?
The formula for finding the radius of a small circle is r = d/2, where r is the radius and d is the diameter of the circle. The diameter is the distance across the circle passing through the center.
Can you find the radius of a small circle if you only know the circumference?
Yes, you can find the radius of a small circle if you know the circumference. The formula for this is r = C/2π, where r is the radius and C is the circumference of the circle. π is a constant value of approximately 3.14.
How can you find the radius of a small circle if you know the area?
To find the radius of a small circle if you know the area, you can use the formula r = √(A/π), where r is the radius and A is the area of the circle. Again, π is a constant value of approximately 3.14.
Is the radius of a small circle always the same as the diameter?
No, the radius and diameter of a small circle are not always the same. The diameter is always twice the length of the radius, but they can have different numerical values depending on the size of the circle.