Chapter 9.2, Problem 17WE

To determine

To Find :The type of given quadrilateral JPQK.

Answer to Problem 17WE

The given quadrilateralJPQK must be a Trapezium.

Explanation of Solution

Given information:

  

JK is tangent to the circle with centers P and Q.

  

The measures of the sides are,

  $\begin{array}{l}PQ=17\hfill \\ QK=3\hfill \\ PD=3\hfill \\ PJ=11\hfill \end{array}$

OT is required to find the measures of sides JD and JK.

Now,

  $PJ\perp JK$ And $QK\perp JK$

Joint PJ and QK and make a line parallel to PQ that is DK.

Then, it implies that $PQ=DK\left(=17\right)$ ........(1)

To find DK;

Triangle JKD is a right angle triangle so one can use Pythagoras Theorem,

The measures of sides in this right angle triangle are,

  $\begin{array}{l}DK=17\hfill \\ JD=11–3\hfill \\ =8\hfill \end{array}$

Use Pythagoras Theorem as follows;

  $\begin{array}{l}DK²=JK²+JD²\hfill \\ 17²=JK²+8²\hfill \\ 289=JK²+64\hfill \\ JK²=289–64\hfill \\ =225\hfill \end{array}$

Take positive square to get,

JK= 15

Therefore, the required length is JK=15.

In quadrilateral PQJK, sum one pair of adjacent angle is $180°$ .

So it can be concluded that the opposite sides is parallel.