Question

1. if \\(\overline{lk} \cong \overline{mk}\\), \\(lk = 7x - 10\\), \\(kn = x + 3\\), \\(mn = 9x - 11\\), and \\(kj = 28\\), find \\(lj\\).

Explanation:

Step1: Use \( \overline{LK} \cong \overline{MK} \) to find \( x \)

Since \( \overline{LK} \cong \overline{MK} \), their lengths are equal. Also, \( MN = MK + KN \), and \( MK = LK = 7x - 10 \), \( KN = x + 3 \), \( MN = 9x - 11 \). So we can set up the equation:
\( 7x - 10 + x + 3 = 9x - 11 \)
Simplify left side: \( 8x - 7 = 9x - 11 \)
Subtract \( 8x \) from both sides: \( -7 = x - 11 \)
Add 11 to both sides: \( x = 4 \)

Step2: Find length of \( LK \)

Substitute \( x = 4 \) into \( LK = 7x - 10 \):
\( LK = 7(4) - 10 = 28 - 10 = 18 \)

Step3: Find length of \( LJ \)

\( LJ = LK + KJ \), we know \( LK = 18 \) and \( KJ = 28 \), so:
\( LJ = 18 + 28 = 46 \)

Answer:

\( 46 \)