Question

find the pero bisector of mn find the value of x and the perimeter of MNG

Answer 1

Since HG is the perpendicular bisector of MN, NH=HM; thus, HM=8. Furthermore,

$\begin{gathered} \angle NHG=90degree \\ \text{and} \\ \angle MHG=90degree \end{gathered}$

Thus, NHG and MHG are right triangles that have two sides of equal length

$\begin{gathered} HG=HG \\ \text{and} \\ NH=HM \end{gathered}$

Therefore,

$\Rightarrow\Delta\text{HGN}\cong\Delta HGM$

Finally,

$\begin{gathered} \Rightarrow GN=GM \\ \Rightarrow2x=21-x \\ \Rightarrow3x=21 \\ \Rightarrow x=7 \end{gathered}$

And the perimeter of triangle MNG is

$\begin{gathered} P=NG+NM+MG=NG+(NH+HM)+MG=2x+(8+8)+21-x \\ \Rightarrow P=x+37 \\ \Rightarrow P=7+37=44 \end{gathered}$

Therefore, the answers are x=7, and the Perimeter of MNG is equal to 44