Chapter 7, Problem 15AE

To determine

To find: the values of $x$ and $y.$

Answer to Problem 15AE

  $x=6$ and $y=\frac{7}{2}=3.5$

Explanation of Solution

Given:

  $AB$$\parallel$$DC$

  $AD$$\parallel$$GC$

  

  Since $AD$$\parallel$$GC$ , so $AD$$\parallel$$BC.$

  $\Rightarrow AB=DC$ (distance between parallel lines is equal throughout)

   Since, $AB$$\parallel$$DC$ , so

  $\Rightarrow AD=BC$ (distance between parallel lines is equal throughout)

   In $\Delta DEC$ and $\Delta AEF,$

  $\angle DEC=\angle AEF$ (vertically opposite angles)

  $\angle EDC=\angle EFA$ (alternate interior angles are equal)

  $\therefore \Delta DEC$ is similar to $\Delta AEF$ by Angle-Angle (AA) similarity criteria.

So,

  $\frac{DC}{AF}=\frac{AE}{EC}$

  $\Rightarrow \frac{12}{8}=\frac{x}{9}$

  $\Rightarrow x=\frac{8\times 9}{12}=6$

  $\Rightarrow x=6$

In $\Delta DEA$ and $\Delta CEG,$

  $\angle DEA=\angle CEG$ (vertically opposite angles)

  $\angle EAD=\angle ECG$ (alternate interior angles are equal)

  $\therefore \Delta DEA$ is similar to $\Delta CEG$ by Angle-Angle (AA) similarity criteria.

So,

  $\frac{AE}{EC}=\frac{AD}{CG}$

  $\Rightarrow \frac{x}{9}=\frac{7}{7+y}$

  $\Rightarrow 7+y=\frac{7\times 9}{6}$ ( $\because x=6$ )

  $\Rightarrow y=\frac{7}{2}=3.5$

Conclusion:

Hence, the values are $x=6$ and $y=\frac{7}{2}=3.5$ .