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 congruent segments

Since \(\overline{LK} \cong \overline{MK}\), we have \(LK = MK\). Also, \(MN = MK + KN\), so substitute \(MK\) with \(LK\) (which is \(7x - 10\)) and \(KN\) with \(x + 3\). So \(MN=(7x - 10)+(x + 3)\). We know \(MN = 9x - 11\), so set up the equation:
\[9x - 11=(7x - 10)+(x + 3)\]

Step2: Simplify the equation

Simplify the right - hand side: \((7x - 10)+(x + 3)=7x+x-10 + 3=8x-7\). So the equation becomes:
\[9x - 11=8x - 7\]
Subtract \(8x\) from both sides: \(9x-8x-11=8x - 8x-7\), which gives \(x - 11=-7\). Then add 11 to both sides: \(x-11 + 11=-7 + 11\), so \(x = 4\).

Step3: Find the length of \(LK\)

Substitute \(x = 4\) into the expression for \(LK\): \(LK=7x-10\). So \(LK = 7\times4-10=28 - 10 = 18\).

Step4: Find the length of \(LJ\)

We know that \(LJ=LK + KJ\). We found \(LK = 18\) and \(KJ = 28\). So \(LJ=18 + 28=46\).

Answer:

\(46\)