Geometry Questions About Central Angles and Secants?

Question

I have 2 problems that I can't figure out and my parents don't remember how to do. Please help me!! grade depends on it!!

1) Circle O has a central angle measure of X. Minor Arc DF has a measure of 4x-30. What is X?

2) PAB is a secant to circle O and meets at point A. PCD is secant to circle O and meets it at point C. PA= 4. PB= 18. PD=24. What is the length of PC?

Please help me. I would REALLY appreciate it.

Answer 1

The central angle and minor arc are both parts of the whole circle. If you set up a proportion for the angle and the circumference, the values should be the same. So with what you're given, you have:

X/360 = (4x - 30)/(2*pi*r)

At this point, there's too much you don't know - you need to know the radius or the circumference of the circle to finish this out. Otherwise, this is as far as you can get. Even if we assume that X and x are the same (a typo perhaps?), the best you can do is solve x in terms of r:

x/360 = (4x - 30)/(2*pi*r)
2*pi*r*x = 360(4x - 30)
pi*rx = 720x - 5400
pi*r + 5400 = 719x
x = (pi*r + 5400)/719

...and that doesn't look to pretty...Double check your figure and the problem and see if you can use any of the above to help with it.

As for your second one, there is a theorem that goes something like PA * PB = PC * PD. Since you have 3 of the 4 values, we can plug in and find the remaining one, PC:

PA * PB = PC * PD
4 * 18 = PC * 24
72 = PC * 24
PC = 3

Answer 2

there is not something specific approximately those angles, yet there's a theorem, asserting : the perspective between to secants, intersecting in a circle is comparable to a million/2 the sum of the inscribed angles. If the secants intersect exterior the circle then the perspective is a million/2 the adaptation of the inscribed angles. a undeniable case is while 2 secants intersect on the circle, then the perspective is a million/2 of the inscribed perspective. (if inscribed perspective is the suited notice to indicate the arcs of the circles between 2 factors, english isn't my interior of sight language)