Line segment XY is tangent to circle Z at point U.
If the measure of UV is 84º, what is the measure of ZYUV?

Step-by-step explanation:

The picture of the question in the attached figure

we have that

-----> given problem

so

------> by central angle

we know that

The triangle UZV is an isosceles triangle

because

ZU=ZV=radius

so

-----> bases angle of the isosceles triangle

Remember that

The sum of the internal angles of any triangle must be equal to 180 degrees

so

substitute and solve for m<ZUV

Find the measure of angle YUV

we have that

------> by complementary angles

substitute the given value and solve for m<YUV

the answer

b. they are parallel and will never intersect.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4