Question

line m is tangent to the circle. find mrst. mrst = °

Explanation:

Step1: Use the tangent - arc theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Let the intercepted arc be \(x\). The angle between the tangent \(m\) and the chord \(TR\) is \(98^{\circ}\). The formula is \(\text{Angle}=\frac{1}{2}\times\text{Intercepted arc}\). But here, the angle between the tangent and the chord is \(98^{\circ}\), and the measure of the major arc \(RST\) can be found using the fact that the measure of the angle formed by a tangent and a chord is half the measure of the intercepted arc. The intercepted arc for the \(98^{\circ}\) angle is the minor arc \(RT\). Let \(m\overset{\frown}{RT}=y\). Then \(98^{\circ}=\frac{1}{2}y\), so \(y = 196^{\circ}\) is wrong. Wait, another approach: The sum of the measures of a major arc and a minor arc is \(360^{\circ}\). The measure of the angle between the tangent and the chord is half the measure of the intercepted arc. The angle between the tangent \(m\) and chord \(TR\) is \(98^{\circ}\), and the intercepted arc for the angle (using the formula \(\text{Angle}=\frac{1}{2}\times\text{Intercepted arc}\)): If the angle between tangent and chord is \(\theta\), and the intercepted arc is \(A\), \(\theta=\frac{1}{2}A\). But for the major arc \(RST\), we know that the sum of the measures of arcs in a circle is \(360^{\circ}\). Also, the measure of the angle between the tangent \(m\) and chord \(TR\) is half the measure of the major arc \(RST\) (by the tangent - arc theorem: \(\text{Angle formed by tangent and chord}=\frac{1}{2}\times\text{measure of intercepted arc}\)).

Step2: Calculate \(m\overset{\frown}{RST}\)

Since the angle between the tangent \(m\) and chord \(TR\) is \(98^{\circ}\), and by the tangent - arc theorem \(\text{Angle}=\frac{1}{2}\times\text{Intercepted arc}\). Let \(m\overset{\frown}{RST}=x\). Then \(98^{\circ}=\frac{1}{2}x\). Multiply both sides by \(2\): \(x = 196^{\circ}\)

Answer:

\(196\)