Question

if gj = ij = 32, gh = 3w, and hi = w + 36, what is the value of w?

w = \boxed{}

Explanation:

Step1: Use the perpendicular bisector theorem

Since \( HJ \) is the perpendicular bisector of \( GI \) (given \( GJ = IJ = 32 \)), by the perpendicular bisector theorem, \( GH=HI \).

Step2: Set up the equation

We know \( GH = 3w \) and \( HI=w + 36 \). Since \( GH = HI \), we have the equation \( 3w=w + 36 \).

Step3: Solve the equation for \( w \)

Subtract \( w \) from both sides: \( 3w-w=w + 36-w \).
This simplifies to \( 2w=36 \).
Divide both sides by 2: \( w=\frac{36}{2} \).

Answer:

\( 18 \)