Question

if ij = jk = 96, hi = 11a, and hk = 13a - 16, what is the value of a?
(there is a diagram with points i, h, k and j, where hj is a horizontal line with a right angle at h to line ik, forming a geometric figure.)
a = \boxed{}

Explanation:

Step1: Apply the perpendicular bisector theorem

Since \( HJ\) is a perpendicular bisector of \( IK\) (given the right - angle symbol at \( H\)), by the perpendicular bisector theorem, \( HI = HK\).

Step2: Set up the equation

We know that \( HI = 11a\) and \( HK=13a - 16\). So, \(11a=13a - 16\).

Step3: Solve the equation for \(a\)

Subtract \(11a\) from both sides: \(0 = 13a-11a - 16\), which simplifies to \(0 = 2a-16\).
Add \(16\) to both sides: \(16 = 2a\).
Divide both sides by \(2\): \(a=\frac{16}{2}\).

Answer:

\(8\)