Question

I'm not fully understanding this and need help.

Points A and B lie on a circle centered at point O. If length of own AB/radius=π/10, what is the ratio of the area of sector AOB to the area of the circle?
A.) 1/10
B.) π/10
C.) 1/20
D.) π/20

I already know B isn't the answer. I'm thinking A is.

Answer 1

when you say "own AB" do you mean "arc AB" ?

Answer 2

I believe so

Answer 3

there is an expression
\[ r \theta = s\]
which means radius times the angle (in radians) = the length of the arc

Answer 4

for your problem, where we use arc AB instead of "s"
\[ r \theta = arc AB \]
if we divide both sides by r we get
\[ \theta = \frac{arc\ AB}{r} \]
the question says that is pi/10
\[ \theta = \frac{arc\ AB}{r} = \frac{\pi}{10}\]

Answer 5

I'm following

Answer 6

I think you want the ratio of theta (the angle of the sector) divided by 2pi (the angle of the whole circle)

Answer 7

or instead of dividing theta by 2pi, multiply by 1/(2pi)

Answer 8

the idea is the angle of sector / 2pi is the fraction of the whole circle we want to know
\[ \frac{\pi}{10} \cdot \frac{1}{2 \pi} \]

Answer 9

So I would solve that?

Answer 10

strictly speaking you only solve equations (which have = signs)
in this case, you simplify.
notice you have a pi "up top" and "down below" which means pi/pi = 1 (i.e. it cancels)
to multiply the fraction, you multiply top time top and bottom times bottom
(and of course simplify pi/pi to 1)

Answer 11

So replace both pi with 1
(I'm trying to fully understand ^^;)

Answer 12

it works the same way as with numbers. for example if you had
6/3 (which we know is 2)
but we write it as
\[ \frac{3\cdot 2}{3} = \frac{3}{3} \cdot 2 = 1 \cdot 2 = 2\]

Answer 13

think of the pi as the "3" in that example
\[ \frac{\pi}{10} \cdot \frac{1}{2 \pi} = \frac{\pi}{\pi} \cdot \frac{1}{10 \cdot 2} \]

Answer 14

2pi means 2 * pi
so the top is 1*pi the bottom is 2*10*pi and the pi's cancel

Answer 15

Oh I understand it now.
The pi would cancel out and then the answer would pretty much be right in front of you. (As you just said)

Answer 16

the top simplifies to 1
the bottom to 2*10 which (I assume) you know is 20

Answer 17

Thank you so much

Answer 18

yw