Question

Line segment XY is tangent to circle Z at point U. If the measure of UV is 84, what is the measure of YUV. A. 42 B. 84 C. 96 D. 168

Answer 1

Answer is 42. Don't listen to the guy that said it's 168. I did the test and it marked it wrong.

Answer 2

Answer:

The measure of angle YUV is equal to $m<YUV=42\°$

Step-by-step explanation:

see the attached figure to better understand the problem

we have that

$arc\ UV=84\°$ -----> given problem

so

$m<UZV=84\°$ ------> by central angle

we know that

The triangle UZV is an isosceles triangle

because

$ZU=ZV=radius$

so

$m<ZUV=m<ZVU$ -----> bases angle of the isosceles triangle

Remember that

The sum of the internal angles of a triangle is equal to $180\°$

so

$m<ZUV+m<ZVU+m<UZV=180\°$

$2m<ZUV+m<UZV=180\°$

substitute and solve for m<ZUV

$2m<ZUV+84\°=180\°$

$m<ZUV=(180\°-84\°)/2=48\°$

$m<ZUV+m<YUV=90\°$ ------> by complementary angles

solve for m<YUZ

$48\°+m<YUV=90\°$

$m<YUV=90\°-48\°=42\°$