Hello! Any help with these problems is greatly appreciated!
A circle is centered at O. The tangent to the circle at P is extended to Q. Line segment QS intersects the circle at R Given that. OS = 2, SR = RQ = 3 and PQ = 6, find the radius of the circle.
Let AB be a diameter of a circle, and let C be a point on the circle such that AC=8 and BC=6. The angle bisector of angle ACB intersects the circle at point M. Find CM.
Thank you so much!!
Hello fellow AoPSer!
For the first one you should extend QS such that it intersects the circle again at T. And extend OS to intersect the circle at U and V. Try to find ST from there and set variables to solve an equation with power of a point.
For the second one. Let the intersection of AB and MC be D. Drop an altitude from D to AC. Then use the Angle Bisector Theorem, Similar Triangles, and lastly Power of a Point to get your answer.
Hope this helped!
For the first one:
1) Extend QRS to point T on the circle.
2) By the "power of a point": QP2 = QR·QT ---> 62 = 3·QT ---> QT = 12
Since QR = 3, RT = 9.
Since RS = 3, ST = 6.
3) Draw OT and OR, creating triangle(TSO) and triangle(RSO).
OT and OR are radii; Let the radius = r.
4) Use the Law of Cosines on these two triangles.
OT2 = r2 = 62 + 22 - 2·6·2·cos(TSO)
OR2 = r2 = 32 + 22 - 2·3·2·cos(RSO)
Since these are equal: 62 + 22 - 2·6·2·cos(TSO) = 32 + 22 - 2·3·2·cos(RSO)
40 - 24·cos(TSO) = 13 - 12·cos(RSO)
Since angle(RSO) is supplementary to angle(TSO), cos(RSO) = - cos(TSO)
---> 40 - 24·cos(TSO) = 13 + 12·cos(TSO)
27 = 36·cos(TSO)
0.75 = cos(TSO)
5) Since OT2 = r2 = 62 + 22 - 2·6·2·cos(TSO)
---> r2 = 40 - 24·0.75
r2 = 22
r = sqrt(22)
2) Let's place a point P on a circle opposite to point M. Now we have a right triangle CMP
∠ABC = arctan( AC / BC )
∠CMP = ∠ABC - ∠BCM
CM = cos(∠CMP) * MP
1) Extend line segment QS to point T on a circle.
QT = PQ2 / RQ RT = QT - RQ
M is a midpoint of line segment RT. Now we have a right triangle OMS.
MO = sqrt( OS2 - MS2 )
Radius OR = sqrt( MR2 + MO2 )
Can I have help with the steps for the second one? Could someone explain it?