Question

if $overline{eh}congoverline{gh}$, $ef = z + 80$, and $fg = 5z + 32$, what is the value of $z$?

Explanation:

Step1: Use congruent - segment property

Since $\overline{EH}\cong\overline{GH}$, by the property of the perpendicular bisector (a point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment), we have $EF = FG$.
So, $z + 80=5z+32$.

Step2: Solve the equation for $z$

Subtract $z$ from both sides: $80 = 5z - z+32$, which simplifies to $80 = 4z+32$.
Then subtract 32 from both sides: $80 - 32=4z$, so $48 = 4z$.
Divide both sides by 4: $z=\frac{48}{4}=12$.

Answer:

$12$