SOLUTION: Chord PQ is perpendicular bisector of radius OA of a circle with center O (A is a point on the edge of the circle). If the length of the arc PAQ = 2pie/3. What is the length of the

Algebra ->  Circles -> SOLUTION: Chord PQ is perpendicular bisector of radius OA of a circle with center O (A is a point on the edge of the circle). If the length of the arc PAQ = 2pie/3. What is the length of the      Log On


   

Question 1085256: Chord PQ is perpendicular bisector of radius OA of a circle with center O (A is a point on the edge of the circle). If the length of the arc PAQ = 2pie/3. What is the length of the chord PQ?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Here is a sketch, with the midpoint of OA labeled as R:
Take a good look at right triangle OQR.
The lengths OP=OA=OQ=R are the radius, R , of the circle.
OR=RA=%281%2F2%29OA=%281%2F2%29R=%281%2F2%29OQ is half of that radius.
That tells you that the short leg of right triangle OQR is half the hypotenuse.
It means that
RQ=sqrt%283%29R%2F2 (from applying the Pythagorean theorem).
It also means (considering trigonometric ratios for the angles of OQR) that
OQR is a 30-60-90 triangle, with a 60%5Eo angle ROQ=AOQ at O,
which makes angle POQ and arc PAQ measure 2%2A60%5Eo=120%5Eo ,
or 1%2F3 of the whole circle.
If 2pi%2F3 is the length of arc PAQ ,
then 3%282pi%2F3%29=2pi is 2pi%2AR , the length of the circumference.
Then, R=1 , RQ=sqrt%283%29%2A1%2F2=sqrt%283%29%2F2 , and PQ=2RQ=2%28sqrt%283%29%2F2%29=highlight%28sqrt%283%29=about1.732%29 .