A quadrilateral circumscribed about a circle has angles of 80,90,94, and 96. Find the measures of the four arcs determined by the points of tangency.80,90,94, and 96 are in degrees.
Let the angles be `/_A=80^@,/_B=90^@,/_C=94^@,/_D=96^@` , and let the points of tangency be x(between A and B), y,z,w in order.
The measure of an angle formed by two tangents is equal to 1/2 the difference of the two arcs. Thus:
Let a=arc(xy),b=arc(yz),c=arc(wz) and d=arc(wx)
Then substituting we get:
We know that a=90. (A radius drawn to a point of tangency is perpendicular to the tangent; thus since angle B is 90 and the two radii to the points of tangent are 90 then the central angle is 90 and thus the arc is 90)
Then we have the following system:
Adding the first to the second yields 340=2b+2c or 170=b+c
Adding the third to the fourth yields 200=2d or d=100
Adding the second to the third yields 368=2c+2d or 2c=168; c=84
Thus a=90,b=86,c=84, and d=100. These are the four arcs.