Question 1083220: Circles M and K are congruent, arc QR is congruent to arc LN and arc OP is congruent to
arc VW . Find x and y.
circle M with chord QR, measuring 8 and chord WV measuring 2 x, circle K with chord LN measuring 3 x plus 4 y and chord PO measuring 5 y minus 3
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! This problem has enough letters (K, L, M, N O, P, Q, R) to make letter soup.
To make matters worse, the letter orders are sometimes swapped, for example
calling a chord PO, when it could have been called chord OP, just like the corresponding arc OP.
Do not let any of that confuse you.
A sketch helps.
Here is circle M, with arc QR and chord QR.
As circles are often named with the name of their center,
the center of this circle is point M.
Circles K is congruent to circle M,
and arc LN is congruent to arc QR.
That means that we can move circle K to the point where both circles are exactly superimposed,
and do it in such way that arc QR and arc LN are superimposed.
of course, then chords QR and LN would be exactly superimposed.
That proves that they are congruent,
and that means they have the same length.
If congruent arcs are on congruent circles,
the corresponding chords are congruent.
Maybe that is a theorem in your geometry textbook,
but it is so easy to see that you do not need to memorize it.
So,  because chords LN and QR have the same length.
Similarly,
 because chords PO and WV have the same length.
The whole letter soup (pun intended) boils down to
and .
-->  -->  --> 
Substituting the value found for  ,
-->  -->  -->  -->  -->  --> 
|
|
|
