Chapter 3.3, Problem 12PSB

To determine

  $Tofindthemeasureofm\angle GOH,m\angle GHOandGH.$

Answer to Problem 12PSB

  $Themeasureofm\angle GOH=67,m\angle GHO=25andGH=13.5$

Explanation of Solution

Given information:

  $\begin{array}{l}Histhemidpointof\overline{GJ}.\\ Misthemidpointof\overline{OK}.\end{array}$

  $\begin{array}{l}\overline{GO}=\overline{JK},\\ \overline{GJ}=\overline{OK},\\ \angle G\cong \angle J,\\ OK=27,\\ m\angle GOH=x+24,\\ m\angle GHO=2y-7,\\ m\angle JMK=3y-23,\\ m\angle MJK=4x-105.\end{array}$

Formula used:

As corresponding parts of congruent triangles are congruent,

  $m\angle GOK\cong m\angle MJK$

  $m\angle GHO\cong m\angle JMK$

Calculation:

  

By definition of mid point,

  $\overline{GH}\cong \overline{MK}$

By SAS congruence rule,

  $\Delta GHO\cong \Delta KMJ$

  $\begin{array}{l}\begin{array}{l}OK=27,\\ MK=(1/2)OK\\ \therefore MK=13.5\\ \overline{GH}\cong \overline{MK}\end{array}\hfill \\ \therefore GH=13.5\hfill \\ \begin{array}{l}m\angle GOK\cong m\angle MJK\\ x+24=4x-105\\ \therefore -3x=-129\\ \therefore x=43\\ m\angle GOH=x+24\\ =43+24\\ =67\\ m\angle GHO\cong m\angle JMK\\ 2y-7=3y-23\\ \therefore y=16\\ m\angle GHO=2y-7\\ =(2*16)-7\\ =25\end{array}\hfill \end{array}$