Answer:
o find MW, we need to use the fact that OS is perpendicular to WV, which means that OS is also perpendicular to MW since it bisects WV.
Let's label the midpoint of WV as point N. Then we can use the Pythagorean theorem to find MW.
First, we need to find the length of ON. Since OS is a radius of the circle, it is equal to the radius of the circle, which we can call r. Then, using the Pythagorean theorem, we have:
ON^2 = OS^2 - SN^2
ON^2 = r^2 - (WV/2)^2
ON^2 = r^2 - (MW/2)^2 (since NW = MV)
Next, we need to find the length of MN. We know that OM is half of WV, so OM = WV/2. Then, using the Pythagorean theorem again, we have:
MN^2 = ON^2 + OM^2
MN^2 = r^2 - (MW/2)^2 + (WV/2)^2
MN^2 = r^2 - (MW/2)^2 + (2MW/2)^2 (since WV = 2MW)
MN^2 = r^2 - (MW/2)^2 + MW^2
Finally, we can solve for MW by using the Pythagorean theorem one more time:
MW^2 = MN^2 + NW^2
MW^2 = (r^2 - (MW/2)^2 + MW^2) + (MW/2)^2
MW^2 = r^2 - (MW/2)^2 + MW^2/4 + MW^2/4
MW^2 = r^2 - (MW/2)^2 + MW^2/2
Multiplying both sides by 4 gives:
4MW^2 = 4r^2 - MW^2 + 2MW^2
3MW^2 = 4r^2
MW^2 = 4r^2/3
MW = 2r/sqrt(3)
Therefore, the length of MW is 2r/sqrt(3).