Angles and Segments in Circles

ANSWER:

This looks like the case where the intersection is inside of the circle. This means the segments are related by their products. This gives us

$\begin{aligned} (2.1)(6.2) &= 2.8x \\ 13.02 &= 2.8x \\ 4.65 &= x \end{aligned}$

This gives us the value of x as 6.25.

ANSWER:

Because we have a right angle with the circle we know that $\overleftrightarrow{DC}$ is tangent to the circle. From this we have the following relationship.

$\overline{DC}^2 = \overline{CA} \times \overline{CB}$

From this we can substitute our given information

$\begin{aligned} \overline{DC}^2 &= 52 \times 28 \\ \overline{DC}^2 &= 1456 \\ \overline{DC} &= \sqrt{1456} \\ \overline{DC} &\approx 38.16 \end{aligned}$

With the value of this side of a right triangle, we can then use the Pythagorean Theorem to find $\overline{AD}$.

$\begin{aligned} \overline{DA}^2 + \overline{DC}^2 &= \overline{AC}^2 \\ \overline{DA}^2 + \sqrt{1456}^2 &= 52^2 \\ \overline{DA}^2 + 1456 &= 2704 \\ \overline{DA}^2 &= 1248 \\ \overline{DA} &= \sqrt{1248} \\ \overline{DA} &\approx 35.33 \end{aligned}$

This length is a diameter, so for the last step we simply need to divide by two to find the radius of the circle.

This gives us that the radius is about 17.67 feet.