Question

$\overline{bc}$ is tangent to circle a and $\overline{ab}$ is a radius.
what is the measure of $\angle b$? \\(\text{select}\\) degrees
what is the length of $\overline{ac}$? $ac = \text{select}$ cm

Explanation:

For the measure of ∠B:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. Since \(\overline{BC}\) is tangent to Circle A at point B and \(\overline{AB}\) is a radius, \(\angle B = 90^{\circ}\) (because the radius is perpendicular to the tangent at the point of contact).

For the length of \(\overline{AC}\):

Step1: Identify triangle type

Since \(\angle B = 90^{\circ}\), \(\triangle ABC\) is a right - triangle with \(AB = 6\space cm\) and \(BC=8\space cm\), and we want to find the length of the hypotenuse \(AC\).

Step2: Apply Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). In \(\triangle ABC\), let \(a = AB = 6\space cm\), \(b = BC = 8\space cm\) and \(c = AC\). Then \(AC^{2}=AB^{2}+BC^{2}\).
Substitute the values: \(AC^{2}=6^{2}+8^{2}=36 + 64=100\).

Step3: Solve for \(AC\)

Take the square root of both sides: \(AC=\sqrt{100} = 10\space cm\).

Answer:

for ∠B:
\(90\) degrees