Question

a secant and a tangent meet at a 90° angle outside the circle. what must be the difference between the measures of the intercepted arcs? 90° 270° 45° 180°

Explanation:

Step1: Recall the formula for angle outside a circle

The measure of an angle formed by a secant and a tangent outside a circle is half the difference of the measures of the intercepted arcs. The formula is: $m\angle = \frac{1}{2}(m\overset{\frown}{major} - m\overset{\frown}{minor})$, where $m\angle$ is the measure of the angle outside the circle, $m\overset{\frown}{major}$ is the measure of the major intercepted arc, and $m\overset{\frown}{minor}$ is the measure of the minor intercepted arc.

Step2: Substitute the given angle measure

We are given that the angle outside the circle is $90^{\circ}$. Substituting into the formula: $90^{\circ}=\frac{1}{2}(m\overset{\frown}{major} - m\overset{\frown}{minor})$.

Step3: Solve for the difference of the arcs

Multiply both sides of the equation by 2 to solve for $(m\overset{\frown}{major} - m\overset{\frown}{minor})$: $2\times90^{\circ}=m\overset{\frown}{major} - m\overset{\frown}{minor}$, so $m\overset{\frown}{major} - m\overset{\frown}{minor}=180^{\circ}$.

Answer:

180°