SOLUTION: If a secant and a tangent intersect in the exterior of the circle to form an 18 degree angle and the lines intercept two arcs on the given circle such that one of them is four thir

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Question 1205674: If a secant and a tangent intersect in the exterior of the circle to form an 18 degree angle and the lines intercept two arcs on the given circle such that one of them is four third of the other, find the measure of each of the intercepted arcs
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.
If a secant and a tangent intersect in the exterior of the circle to form an 18 degree angle
and the lines intercept two arcs on the given circle such that one of them is
four third of the other, find the measure of each of the intercepted arcs.
~~~~~~~~~~~~~~~~~~~~~~~~ 

Let the arcs be alpha and beta.


Then from the problem we have these two equations

    beta = %284%2F3%29%2Aalpha,    (1)

    18 = %281%2F2%29%2A%28beta+-+alpha%29.    (2)


Multiply equation (2) by 2 (both sides) and substitute expression (1) for beta there.
You will get then

    36 = %284%2F3%29%2Aalpha+-+alpha,

    36 = %281%2F3%29%2Aalpha

    36*3 = alpha

    alpha = 108  degrees.


Hence, beta = %284%2F3%29%2Aalpha = %284%2F3%29%2A108 = 4*36 = 144.


ANSWER.  The arcs are 108 degrees and 144 degrees.

Solved.



Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The operative principle you want to learn here is that the measure of the angle formed by two secants and/or tangents to a circle is half the difference between the measures of the two arcs of the circle they cut off.

Then do the calculations as shown by the other tutor.