circle problems need help

Apr 2010
in the accompanying diagram of circle o, ABD and AEC are secants chord BE and CD intersect at F tangents Gh intersect at F tangents Gh intersect circle o at C, mBD = 100, mDE = 70 and mEC = 80



also this problem

if PR = 10 Pq = 6 and PS = 5 find PT

if PR = 12 Pq = 3 and PS = 2 find PT

and this one

in the accompanying diagram of circle o mAB = 64 and mAEB = 52

what is the measure of CD
Mar 2009
Birmingham, Alabama.
For the first problem, using the fact that the angle formed by two secants is equal to the measure of the major arc minus the minor arc, all divided by two. But before we do anything there, determine the measure of arc BC:

\(\displaystyle BC=360-(100+80+70)\)
\(\displaystyle BC=110\)

Therefore angle BAC = \(\displaystyle \frac{\110-70}{2}\)
\(\displaystyle BAC=20\)

since angle BDC is formed by 2 chords, it is half the measure of the arc formed by those 2 chords: \(\displaystyle BDC=\frac{\110}{2}\)
BDC = 55

For CFE you will take the average of 80 and a 100, which is 90.

GCE will be half of its intercepted arc, which is 80, therefore GCE will be 40.

AEB is the supplement of BEC, so AEB will be 180 - BEC. Since BEC is just half of 110 which is 55, AEB =180-55=125