Question

In the figure above, assume that any lines that appear to be tangent, are tangent to the circle. If m<ABD = 38° , and the radius of the circle is 12.6 meters, find CB . Please show all work in order to earn full credit.

Answer 1

To find \( CB \) in the given scenario while utilizing the provided information, we will follow these steps:

### Step 1: Understand the setup

Assuming you have a circle with center \( O \) and a tangent line at point \( A \) intersecting the line segment \( CB \). The angle \( \angle ABD = 38^\circ \).

### Step 2: Use the properties of a tangent line

When a tangent touches a circle, the radius to the point of tangency is perpendicular to the tangent line. Thus, we have:

\[
\angle OAB = 90^\circ
\]

### Step 3: Use the concept of angles

Since we know:

\[
\angle ABD = 38^\circ
\]

we can calculate \( \angle OBD \):

\[
\angle OBD = \angle OAB - \angle ABD = 90^\circ - 38^\circ = 52^\circ
\]

### Step 4: Utilize the law of sines

In triangle \( OBD \),

- We have \( OB \) as the radius, which is \( 12.6 \) m.
- \( OD \) is also a radius of the circle, which is \( 12.6 \) m.
- \( \angle OBD = 52^\circ\).
- \( \angle ODB \) can be calculated knowing that the angles in a triangle add up to \( 180^\circ \):

We know that:
\[
\angle ODB = 180^\circ - \angle OAB - \angle OBD = 180^\circ - 90^\circ - 52^\circ = 38^\circ
\]

### Step 5: Calculating \( CB \)

Using the law of sines in triangle \( OBD \):

\[
\frac{CB}{\sin(\angle OBD)} = \frac{OB}{\sin(\angle ODB)}
\]

Substituting the known values:

\[
\frac{CB}{\sin(52^\circ)} = \frac{12.6}{\sin(38^\circ)}
\]

Now, solving for \( CB \):

\[
CB = \frac{12.6 \cdot \sin(52^\circ)}{\sin(38^\circ)}
\]

### Step 6: Plug in the sine values

Using a calculator gives:

\[
\sin(52^\circ) \approx 0.788
\]
\[
\sin(38^\circ) \approx 0.615
\]

Now substituting these values to find \( CB \):

\[
CB = \frac{12.6 \cdot 0.788}{0.615} \approx \frac{9.91}{0.615} \approx 16.12 \text{ meters}
\]

### Final Answer
CB is approximately \( 16.12 \) meters.