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?
○ 45°
○ 90°
○ 180°
○ 270°

Explanation:

Step1: Recall the theorem 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: $m\angle = \frac{1}{2}(m\overset{\frown}{large} - m\overset{\frown}{small})$, where $m\angle$ is the measure of the angle outside the circle, and $m\overset{\frown}{large}$ and $m\overset{\frown}{small}$ are the measures of the intercepted arcs.

Step2: Substitute the given angle measure

We are given that the angle is $90^{\circ}$. So we substitute into the formula: $90^{\circ}=\frac{1}{2}(m\overset{\frown}{large} - m\overset{\frown}{small})$

Step3: Solve for the difference of the arcs

Multiply both sides of the equation by 2 to solve for $(m\overset{\frown}{large} - m\overset{\frown}{small})$:

$$\begin{align*} 2\times90^{\circ}&=m\overset{\frown}{large} - m\overset{\frown}{small}\\ 180^{\circ}&=m\overset{\frown}{large} - m\overset{\frown}{small} \end{align*}$$

Answer:

180° (The difference between the measures of the intercepted arcs is 180°)