Question

Solve for x, find angle G, and find the perimeter.

Answer 1

Let's begin by identifying key information given to us:

Triangle HGF is equilateral

a.

`\begin{gathered} \angle F=\angle H=\angle G \\ \angle F=(x+32)^(\circ) \\ \angle H=(2x+4)^(\circ) \\ \Rightarrow(x+32)^(\circ)=(2x+4)^(\circ) \\ x+32=2x+4 \\ \text{Put like terms together, we have:} \\ 32-4=2x-x \\ 28=x\Rightarrow x=28 \\ x=28 \end{gathered}`

Since we know that all the angles are equal, we will substitute the value of x into angle F or H. We have:

`\begin{gathered} \angle H=(2x+4)^(\circ) \\ \angle H=2(28)+4=56+4 \\ \angle H=60^(\circ) \\ But,\angle H=\angle G=60^(\circ) \\ \\ \therefore m\angle G=60^(\circ) \end{gathered}`

b.

`\begin{gathered} Perimeter=HG+GF+FH \\ Perimeter=6y+24 \\ HG=3y-7 \\ \text{The three sides are equivalent, thus we have:} \\ 6y+24=3(3y-7) \\ 6y+24=9y-21 \\ \text{Put like terms together, we have:} \\ 9y-6y=24+21 \\ 3y=45 \\ y=(45)/(3)=15 \\ y=15 \\ \\ Perimeter=6(15)+24=90+24=114 \\ \therefore Perimeter=114 \end{gathered}`