Let $\overline{PQ}$, $\overline{RS}$, and $\overline{TU}$ be parallel chords of a circle. The distance between chords $\overline{PQ}$ and $\overline{RS}$ is 4, and the distance between chords $\overline{RS}$ and $\overline{TU}$ is also 4. If $PQ = 78$ and $TU = 50$, then find $RS$.

Answer 1

let the distance from TU to the centre be x
then RS is x+4 units away from the centre, and
PQ is x+8 units away from the centre

draw the common perpendicular to all the chords, and draw radii to the ends of all the chords
You now have right-angled triangles

from chord TU: r^2 = x^2 + 39^2
from chord PQ: r^2 = (x+8)^2 + 25^2

sp (x+8)^2 + 625 = x^2 + 1521
x^2 + 16x + 64 + 625 = x^2 + 1521
16x = 832
x = 52

r^2 = 52^2 + 39^2 = 4225
r = 65

now to the right-triangle from chord RS
let half the chord by y
(x+4)^2 + y^2 = r^2
56^2 + y^2 = 65^2
y^2 = 1089
y = 33

so RS = 2(33) = 66

Answer 2

don't cheat! this is clearly AoPS! I'm Richard Rusczyk, the founder of AoPS and this is unacceptable! use the message boards for help instead!

Answer 3

Bruh you think you tricking anyone? You had one good sentence and all the other 3 have wrong capitalization, and the fact that you sound like some goody-two-shoes 9 year old. If you know it's AoPS, and you aren't here for exposing cheaters, then you must be cheating too.

Answer 4

yeah like srsly
if you dont cheat then how do u know about this place
richard wouldnt waste his time on finding cheaters
also ur making it look like richard himself said those words and he worldnt be so rude
although tbh searching it up is against the honor code...
"You should not search for potential solutions to problems online or elsewhere. (Especially searching for the problem statement!) Using the internet in this way to search for answers is against our Course Policies! Instead, if you need some assistance with a particular problem, we welcome you to ask questions on your course message board. Your instructor and other AoPS staff members will be happy to help."

Answer 5

Remember that if you get stuck on a homework problem, you can always ask on the message board! Click on the pencil icon V in the upper-right corner of the problem, and this will open a box where you can ask your question, which will be posted on the message board. You can also click on the speech bubble icon t, which will bring up any discussions on that problem. Seriously, a student showed me this and I cant believe it! Just sop and pay attention in class.

Answer 6

We aint idiots dont try to trick us. Ur not Richard.

Answer 7

you guys are all wrong

Answer 8

um why is everyone fighting 😀