In triangle FGH, GJ is an angle bisector of ∠G and perpendicular to FH. What is the length of FH?

Solution:

Given that FG = 3x - 8,

FJ = x and GH = 16.

In triangle FGH, GJ is an angle bisector of ∠G and perpendicular to FH.

In triangle FGJ and HGJ

∠FJG ≅ ∠HJG [Right angle]

 GJ ≅ GJ  (common)

∠JGF ≅ ∠JGH [Angle bisector]

We know that

Two angles and the included side of one triangle is congruent to the corresponding sides and angles.

Triangle FGJ ≅ Triangle HGJ [By ASA]

FG ≅ HG [c.p.c.t]

FG = HG

Substituting the values

3x - 8 = 16

By further calculation

3x = 16 + 8

3x = 24

x = 24/3

x = 8

FJ ≅ HJ (c.p.c.t)

FJ = HJ = x

Now we should determine the value of FH

FH = FJ + JH [Segment addition property]

It can be written as

FH = x + x

FH = 2x

FH = 2(8)

FH = 16 units

Therefore, the length of FH is 16 units.

---

In triangle FGH, GJ is an angle bisector of ∠G and perpendicular to FH. What is the length of FH?

Summary:

In triangle FGH, GJ is an angle bisector of ∠G and perpendicular to FH. The length of FH is 16 units.