Question

Help please! Mark all statements that are true:
A. If the measure of theta is 60 degrees, then the arc length is r.
B. If the measure of theta is .5 radians, then the arc length is r/2.
C. The ratio of the arc length to r is always equal to 1/pi.
D. If the ratio fo the arc length to r is 1, then the measure of theta is 1 radian.

Answer 1

for part A, use this formula

\[\Large s = \frac{x}{360}*2*\pi*r\]

s = arc length
x = angle in degrees
r = radius

Answer 2

In this case, x = 60

Answer 3

for part A they're asking

if x = 60, does plugging this in yield s = r ?

Answer 4

Okay! Thank you! But how does that help me to know if s=r?

Answer 5

what do you get when you plug in x = 60 and simplify

Answer 6

1/3 pi x r=s

Answer 7

so the result we get is NOT just 'r' all by itself

so the claim made in choice A is false

Answer 8

Oh I see! That's great, thank you very much!

Answer 9

IF it said that the angle was `1 radian` then the claim would be true

if the angle was `1 radian` then the arc length would be equal to the radius r

formula

s = theta*r

where

s = arc length
theta is in radians (this is very important)
r = radius

Answer 10

Hm. So, which part of the question does that help with? Sorry, this is just a trickily worded question.

Answer 11

that's just extra info, but it will help out with B

Answer 12

Great. Thank you!

Answer 13

hint: 0.5 = 1/2

Answer 14

s = theta*r

for part B, theta = 1/2 radians

Answer 15

So B would be true then?

Answer 16

yes because


s = theta*r
s = (1/2)*r
s = r/2

Answer 17

Great. So, I didn't think C was correct. Am I right in my thinking or not?

Answer 18

why are you thinking that

Answer 19

I don't know, just that it seems as if there is not a distinct correlation. Ha. But I don't really know :)

Answer 20

what do you get when you compute `s/r`

replace the 's' with the right side of the first formula I posted

Answer 21

what terms do you see cancelling out?

Answer 22

the radius?

Answer 23

yes r cancels leaving you with `(2pi*x)/360 = (pi*x)/180`

this is NOT equal to `1/pi`

Answer 24

Great! My intuition was correct, thanks for solidifying that! Okay, for the last one, I put that that was correct. Is that true?

Answer 25

@jim_thompson5910

Answer 26

let's find out

Answer 27

dividing s/r using the second formula given yields this


s/r = (theta*r)/r = theta

so the ratio of the arc length s to the radius r is equal to theta


this only true if theta is in radians

Answer 28

if the ratio is equal to 1, then theta is equal to 1 radian

Answer 29

so yep D is true

Answer 30

You are a lifesaver! Thanks so very much!

Answer 31

I got the question right, thank you! @jim_thompson5910