Chapter 3.8, Problem 6PSA

To determine

The chord GJ bisects $\angle HGK$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{GH}\cong \overrightarrow{GK}$

GJ is an altitude

Formula used : similar angles in congruent triangles are always congruent

Proof : As altitude is always perpendicular to the main line segment, we can say that

  $\angle GJH\cong \angle GJK={90}^0$

Thus, now in triangles GJH and GJK, we apply Pythagoras theorem to find

  $\overrightarrow{JH}\cong \overrightarrow{JK}$

Thus, we know that GJH and GJK are congruent by SSS property. Thus, we know that for congruent triangles,

  $\angle HGJ\cong \angle JGK$

So, this can also be said in a way that

Chord GJ bisects $\angle HGK$

Hence, given details are proved