Question

what is fi?
fi =
submit

Explanation:

Step1: Identify the geometric principle

This is a problem about angle bisectors and perpendiculars. The angle bisector theorem for distances: if a point (H) is on the angle bisector, then it is equidistant from the two sides of the angle. Here, HG is perpendicular to one side (the line with G), and HI is perpendicular to the other side (the line with I). Also, HF is the angle bisector. So, the distance from H to FG (HG) and the distance from H to FI (HI) should be equal? Wait, no, actually, FG and FI? Wait, looking at the diagram, FG is length 8, and HG is perpendicular to FG, HI is perpendicular to FI. Since H is on the angle bisector, the lengths of the perpendiculars from H to the two sides are equal, but also, in this case, since FG and FI are segments from F to the perpendiculars, and H is on the angle bisector, by the angle - bisector property, FG = FI.

Step2: Apply the angle - bisector property

We know that FG = 8. By the property of the angle bisector (a point on the angle bisector is equidistant from the two sides, and in this case, the segments from F to the perpendiculars on the two sides are equal), we can conclude that FI=FG.

Step3: Calculate the length of FI

Since FG = 8, then FI = 8.

Answer:

8