Question

if △ghj is an equilateral triangle, gh = 5x - 13, hj = 11x - 61, and gj = 7x - 29, find x and the measure of each side

Explanation:

Step1: Set two - side lengths equal

Since $\triangle GHJ$ is equilateral, $GH = HJ$. So we set up the equation $5x - 13=11x - 61$.

Step2: Solve the equation for $x$

Subtract $5x$ from both sides: $- 13 = 11x-5x - 61$, which simplifies to $-13 = 6x - 61$.
Add 61 to both sides: $-13 + 61=6x$, so $48 = 6x$.
Divide both sides by 6: $x=\frac{48}{6}=8$.

Step3: Find the length of each side

Substitute $x = 8$ into the expression for $GH$ (we could use any of the side - length expressions).
$GH=5x - 13=5\times8 - 13=40 - 13 = 27$.
Since it's an equilateral triangle, $HJ = 27$ and $GJ = 27$.

Answer:

$x = 8$, $GH=27$, $HJ = 27$, $GJ = 27$