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 formula for the angle formed outside the circle

The formula for the angle formed by a secant and a tangent outside a circle is $\theta=\frac{1}{2}(m\overset{\frown}{AB}-m\overset{\frown}{CD})$, where $\theta$ is the angle outside the circle, $m\overset{\frown}{AB}$ is the measure of the intercepted major arc, and $m\overset{\frown}{CD}$ is the measure of the intercepted minor arc.

Step2: Substitute the given angle into the formula

Given $\theta = 90^{\circ}$, we substitute into the formula: $90^{\circ}=\frac{1}{2}(m\overset{\frown}{AB}-m\overset{\frown}{CD})$.

Step3: Solve for the difference of the arcs

Multiply both sides of the equation by 2: $2\times90^{\circ}=m\overset{\frown}{AB}-m\overset{\frown}{CD}$.

Answer:

$180^{\circ}$