Question

A circular medallion is hung such that the strings are tangent to the circle as shown in the figure.

image

If the major arc measures 230°, find the degree measure of the angle formed by the strings.

A.
40°

B.
50°

C.
130°

D.
230°

Answer 1

To find the angle formed by the strings that are tangent to the circle, we can use the concept that the angle formed by tangents to a circle from a point outside the circle is half the difference of the measures of the intercepted arcs.

In this case, the major arc measures 230°, leaving the minor arc to measure:

\[
360° - 230° = 130°
\]

Now, the angle formed by the tangents outside the circle, which we'll denote as \( \theta \), is:

\[
\theta = \frac{1}{2} (\text{measure of major arc} - \text{measure of minor arc})
\]

Substituting the known values:

\[
\theta = \frac{1}{2} (230° - 130°)
\]
\[
\theta = \frac{1}{2} (100°)
\]
\[
\theta = 50°
\]

Therefore, the degree measure of the angle formed by the strings is

\[
\boxed{50°}
\]