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 segment addition for MN

Since \( MN = MK + KN \) and \( LK \cong MK \) (so \( MK = LK = 7x - 10 \)), substitute into the equation:
\( 9x - 11=(7x - 10)+(x + 3) \)
Simplify the right - hand side: \( 7x - 10+x + 3=8x - 7 \)
So the equation becomes \( 9x - 11 = 8x - 7 \)

Step2: Solve for x

Subtract \( 8x \) from both sides: \( 9x-8x - 11=8x - 8x-7 \)
Which gives \( x - 11=-7 \)
Add 11 to both sides: \( x-11 + 11=-7 + 11 \)
So \( x = 4 \)

Step3: Find LK

Substitute \( x = 4 \) into \( LK=7x - 10 \):
\( LK=7\times4-10=28 - 10 = 18 \)

Step4: Find LJ

Since \( LJ=LK + KJ \), and \( LK = 18 \), \( KJ = 28 \)
Then \( LJ=18 + 28=46 \)

Answer:

\( 46 \)