1 Attachment(s)

concentric circle problem

I have no idea how to solve this. I love geometry, but I've found that I'm not very naturally good at it. please help!

The circles in the figure I have drawn out are concentric. The chord AB is tangent to the inner circle and has a length of 12 cm. What is the Area of of the non-shaded region? (A of big triangle - A of small triangle)

Attachment 28090

Re: concentric circle problem

Quote:

Originally Posted by

**aaronrpoole** The circles in the figure I have drawn out are concentric. The chord AB is tangent to the inner circle and has a length of 12 cm. What is the Area of of the non-shaded region? (A of big triangle - A of small triangle)

Attachment 28090

As posed, I fear there is no unique answer to this question.

Look at this webpage.

I think that you need to know the radius of one of those two circles. Or some other term from that webpage.

Re: concentric circle problem

Hello, aaronrpoole!

This is a classic problem . . . with a surprising punchline.

Quote:

The circles in the figure I have drawn are concentric.

The chord AB is tangent to the inner circle and has a length of 12 cm.

What is the area of of the non-shaded region? (Area of big circle - Area of small circle)

Code:

` * * *`

* *

* C 6 *

A *- - - - ♥ - - - -♥ B

* | * o

* * r| o* R *

* * ♥ * *

* * O * *

* *

* * *

* *

* *

* * *

$\displaystyle O$ is the center of the circles.

$\displaystyle C$ is the midpoint of chord $\displaystyle AB\!:\;CB = 6$

Let $\displaystyle R = OB$, the radius of the large circle.

Let $\displaystyle r = OC$, the radius of the small circle.

From right triangle $\displaystyle BOC\!:\;r^2 + 6^2 \,=\,R^2 \quad\Rightarrow\quad R^2-r^2 \,=\,36$ .[1]

The area of the large circle is: $\displaystyle \pi R^2$

The area of the small circle is: $\displaystyle \pi r^2$

The area of the ring is: $\displaystyle A \:=\:\pi R^2 - \pi r^2 \:=\:\pi(R^2-r^2)$

Substitute [1]: .$\displaystyle A \:=\:\pi(36) \:=\:36\pi$

Surprise! .We didn't need to know the two radii.

The small circle could be a golfball or the Earth.

The area of the ring is constant!

Re: concentric circle problem

While this subject can be very touchy for most people, my opinion is that there has to be a middle or common ground that we all can find. I do appreciate that youve added relevant and intelligent commentary here though. Thank you!