Question

find the value of ( x ). note that, in the image below, ( x^circ = moverarc{ad} ), ( mangle c = 38.5^circ ), and ( moverarc{bd} = 52^circ ).

Explanation:

Step1: Recall secant-tangent angle theorem

The measure of an angle formed by a tangent and a secant outside a circle is half the difference of the measures of the intercepted arcs. The formula is:
$$m\angle C = \frac{1}{2}(m\overset{\frown}{AD} - m\overset{\frown}{BD})$$

Step2: Substitute known values

Plug $m\angle C=38.5^\circ$, $m\overset{\frown}{AD}=x^\circ$, $m\overset{\frown}{BD}=52^\circ$ into the formula:
$$38.5 = \frac{1}{2}(x - 52)$$

Step3: Solve for $x$

Multiply both sides by 2:
$$77 = x - 52$$
Add 52 to both sides:
$$x = 77 + 52$$

Answer:

$129$