The problem is to find the length of $HJ$ and the measure of angle $GJN$ given certain information about triangle $GHJ$. We are given that $HP = 5x - 16$, $PJ = 3x + 8$, $m\angle GJN = 6y - 3$, $m\angle NJH = 4y + 23$, and $m\angle HMG = 4z + 14$. Also, we know that $GP$ is a median and an angle bisector.

1. Problem Description

The problem is to find the length of $HJ$ and the measure of angle $GJN$ given certain information about triangle $GHJ$. We are given that $HP = 5x - 16$, $PJ = 3x + 8$, $m\angle GJN = 6y - 3$, $m\angle NJH = 4y + 23$, and $m\angle HMG = 4z + 14$. Also, we know that $GP$ is a median and an angle bisector.

2. Solution Steps

Since $GP$ is a median, $P$ is the midpoint of $HJ$. Therefore, $HP = PJ$.

$5x - 16 = 3x + 8$

$2x = 24$

$x = 12$

$HP = 5(12) - 16 = 60 - 16 = 44$

$PJ = 3(12) + 8 = 36 + 8 = 44$

$HJ = HP + PJ = 44 + 44 = 88$

Since $GP$ is an angle bisector, $m\angle GJN = m\angle NJH$.

$6y - 3 = 4y + 23$

$2y = 26$

$y = 13$

$m\angle GJN = 6(13) - 3 = 78 - 3 = 75$

$m\angle NJH = 4(13) + 23 = 52 + 23 = 75$

3. Final Answer

$HJ = 88$

$m\angle GJN = 75$