Question

2. In the circle shown, secants PE and PF have been drawn such that mCE  82 , mDF  98 and the ratio 3. In circle O, tangent MQ , secant MN and chord PN are drawn. If mNPQ  102 and mM  60 ,  COMMON CORE GEOMETRY, UNIT #9 – CIRCLE GEOMETRY - LESSON #7 eMATHINSTRUCTION, RED HOOK, NY 12571, © 2018 Date: _________________  of mCD to mEF is 2:7. Determine the measure of P . Show how you arrived at your answer.

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The arc PR is  $\theta _{PR} = 84$

Step-by-step explanation:

A descriptive diagram of the diagram given in the question is shown on the first uploaded image

  In the circle tangent  $\= {M \= Q}$ , secant $PE$ and  chord $PF$ are drawn

given that  $\theta _ {MPQ} = 102^o$ , $\theta _{M} = 60^o$

   The objective is to determine PR

From the diagram we see that MQ is a straight which implies that the the angle

      $\theta_{NPM} = 180 -102$

      $\theta_{NPM} = 78^o$

Now looking at triangle NMP we see that

          $\theta _{MNP} = 180 -[ \theta_{NMP} + \theta _{NPM}]$

         $\theta _{MNP} = 180 - [60 + 78 ]$

         $\theta _{MNP} = 42 ^o$

Now the measure of an inscribed angle is half the measure of its  arc intercepted, this statement is the inscribed angle theorem

 So with the knowledge

   Then

           $\theta_{RNP} = \frac{1}{2} \theta_{PR}$

  Looking at the diagram we see that

           $\theta _{MNP} = \theta_{RNP} = 42 ^o$

             $42 = \frac{1}{2} \theta _{PR}$

              $\theta _{PR} = 2 * 42$

               $\theta _{PR} = 84$