Question

find hi.
hi =

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that a mid - segment of a triangle (a segment connecting the midpoints of two sides of a triangle) is parallel to the third side and half its length. Here, \(GI\) is a mid - segment of \(\triangle HJK\) (since the red arrows indicate that \(G\) is the midpoint of \(HK\) and \(I\) is the midpoint of \(HJ\)).
The formula based on the mid - segment theorem is \(\frac{HI}{IJ}=\frac{HG}{GK}\). Since \(HG = 18\), \(GK=9\), and \(IJ = 15\). Let \(HI=x\).
We know that \(\frac{x}{15}=\frac{18}{9}\) (because the line segment \(GI\) is parallel to \(KJ\) and by the basic proportionality theorem (Thales' theorem)).

Step2: Solve the proportion

Cross - multiply the proportion \(\frac{x}{15}=\frac{18}{9}\).
We get \(9x=18\times15\).
First, calculate \(18\times15 = 270\). Then, \(x=\frac{270}{9}\).
Since \(270\div9 = 30\).

Answer:

\(30\)