Question

angle bcd is a circumscribed angle of circle a.
what is the length of line segment ac?
10 units
12 units
14 units
16 units

Explanation:

Step1: Identify right triangle

Since $\angle BCD$ is a circumscribed angle, $BC$ is tangent to circle $A$, so $\angle ABC = 90^\circ$. Thus, $\triangle ABC$ is a right triangle with $AB = 8$ (radius), $BC = 6$.

Step2: Apply Pythagorean theorem

In right $\triangle ABC$, $AC^2 = AB^2 + BC^2$.
$$AC = \sqrt{AB^2 + BC^2}$$

Step3: Substitute values and calculate

Substitute $AB=8$, $BC=6$:
$$AC = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$$

Answer:

10 units