Question

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

Answer 1

Answer:

FH = 16 units.

Step-by-step explanation:

Consider the below figure attached with this question.

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

In triangle FGJ and HGJ,

$\angle FJG\cong \angle HJG$             (Right angle)

$GJ\cong GJ$             (Reflexive property)

$\angle JGF\cong \angle JGH$             (Angle bisector)

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

$\triangle FGJ\cong \triangle HGJ$           (By ASA)

Corresponding parts of congruent triangles are congruent.

$FG\cong HG$                     (CPCTC)

$FG=HG$                 (Definition of congruent segment)

$3x-8=16$

$3x=16+8$

$3x=24$

$x=8$

The value of x is 8.

$FJ\cong HJ$                     (CPCTC)

$FJ=HJ=x$                 (Definition of congruent segment)

We need to find the value FH.

$FH=FJ+JH$               (Segment addition property)

$FH=x+x$

$FH=2x$

$FH=2(8)$

$FH=16$

Therefore, the length of FH is 16 units.

Answer 2

If the bisector of an angle in a triangle is perpendicular to the opposite side, the triangle is isosceles. This means FG is congruent to GH because it is an isosceles triangle (a triangle that has two sides of equal length). J bisects FH, but without any additional values, the length of FH cannot be determined.