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 radius is perpendicular to the tangent at the point of tangency. Let the center of the circle be \(O\). If \(PR\) and \(PQ\) are tangents to the circle with center \(O\), then \(\angle{ORP}=\angle{OQP} = 90^{\circ}\). In quadrilateral \(ORPQ\), the sum of interior angles is \(360^{\circ}\). Let the measure of the arc \(RQ\) be \(m\widehat{RQ}\) and the central - angle corresponding to arc \(RQ\) be \(\angle{ROQ}\). The measure of \(\angle{ROQ}=m\widehat{RQ}\) (in degrees, when the arc - measure is in degrees).

Step2: Use angle - sum property of quadrilateral

In quadrilateral \(ORPQ\), we know that \(\angle{ORP}+\angle{OQP}+\angle{ROQ}+\angle{RPQ}=360^{\circ}\). Since \(\angle{ORP} = 90^{\circ}\) and \(\angle{OQP}=90^{\circ}\), we have \(90^{\circ}+90^{\circ}+\angle{ROQ}+\angle{RPQ}=360^{\circ}\).

Step3: Solve for \(\angle{RPQ}\)

Simplify the equation: \(180^{\circ}+\angle{ROQ}+\angle{RPQ}=360^{\circ}\), then \(\angle{RPQ}=180^{\circ}-m\widehat{RQ}\).

Answer:

\(m\angle{RPQ}=180 - m\widehat{RQ}\)