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 by a secant and a tangent outside a circle

The formula for the measure of an 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

We are given that \( \theta = 90^{\circ}\). Substituting into the formula \(90^{\circ}=\frac{1}{2}(m\overset{\frown}{AB}-m\overset{\frown}{CD})\).

Step3: Solve for \(m\overset{\frown}{AB}-m\overset{\frown}{CD}\)

Multiply both sides of the equation \(90^{\circ}=\frac{1}{2}(m\overset{\frown}{AB}-m\overset{\frown}{CD})\) by \(2\). We get \(2\times90^{\circ}=m\overset{\frown}{AB}-m\overset{\frown}{CD}\).

Answer:

\(180^{\circ}\)