Question

consider the figure. use the diagram to verify formulas. part a find the formula for the measure of ∠rpq formed by two tangents to a circle, where the given measure is rq⌢. drag the expressions into the boxes to correctly complete the solution. show hints m∠rpq =

Explanation:

Step1: Recall tangent - circle property

The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs. Let the measure of the major arc be $m\overparen{RQ}$ and the measure of the minor arc be $m\overparen{RQ}$ (assuming the non - reflex and reflex arcs). The formula for $m\angle RPQ$ is $m\angle RPQ=\frac{1}{2}(m\overparen{RQ}-m\overparen{RQ})$ (where one is the major arc and one is the minor arc). If the central angle corresponding to the arc $\overparen{RQ}$ is $\theta$ (in degrees), and the sum of the major and minor arcs of a circle is $360^{\circ}$. Let the minor arc $\overparen{RQ}$ have measure $x$ and the major arc have measure $360 - x$. Then $m\angle RPQ=\frac{1}{2}((360 - x)-x)=\frac{1}{2}(360 - 2x)=180 - x$.

Step2: Use the given arc measure

However, if we assume the given $\overparen{RQ}$ is the minor arc, and let $m\overparen{RQ}=y$, then $m\angle RPQ = 180-\frac{1}{2}m\overparen{RQ}$

Answer:

$m\angle RPQ=180-\frac{1}{2}m\overparen{RQ}$