Question

Circle A is shown. Tangents L M and N M intersect at point M outside of the circle. Arc L N is 270 degrees.
In the diagram of circle A, what is m?

75°
90°
120°
135°

Answer 1

To solve for the measure of angle \( m \) where tangents \( LM \) and \( NM \) intersect outside of circle \( A \), we can use the properties of circle geometry.

Given that arc \( LN \) is 270 degrees, we want to find the measure of angle \( m \) which is the angle formed by the tangents \( LM \) and \( NM \).

The measure of an angle formed by two tangents from a point outside of a circle is equal to half the difference of the measures of the arcs that the tangents subtend. In this case, the angle \( m \) can be computed as follows:

\[
m = \frac{1}{2} \left( \text{measure of arc } LN - \text{measure of arc } L + \text{measure of arc } N \right)
\]

Since arc \( LN \) is 270 degrees, we are interested in the remaining arc measure within the circle.

The full circle is 360 degrees. The remaining arc \( L + N \) is:

\[
360^\circ - 270^\circ = 90^\circ
\]

Now we can apply the formula for angle \( m \):

\[
m = \frac{1}{2} \left( \text{measure of arc } LN - \text{measure of arc } L - \text{measure of arc } N \right) = \frac{1}{2} (270^\circ - 0^\circ) = \frac{1}{2} \cdot 270^\circ = 135^\circ
\]

Therefore, the measure of angle \( m \) is:

\[
\boxed{135^\circ}
\]