Question

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

Explanation:

Step1: Use the congruence property

Since $\overline{LK}\cong\overline{MK}$, we know that $LK = MK$.

Step2: Set up an equation for lengths

We know that $MN=MK + KN$. Substituting the expressions for the lengths: $9x - 11=(7x - 10)+(x + 3)$.

Step3: Simplify the right - hand side of the equation

$(7x - 10)+(x + 3)=7x-10+x + 3=8x-7$. So the equation becomes $9x - 11=8x-7$.

Step4: Solve for $x$

Subtract $8x$ from both sides: $9x-8x-11=8x - 8x-7$, which gives $x-11=-7$. Then add 11 to both sides: $x=-7 + 11=4$.

Step5: Find the length of $LK$

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

Step6: Find the length of $LJ$

We know that $LJ=LK+KJ$. Since $LK = 18$ and $KJ = 28$, then $LJ=18+28=46$.

Answer:

$46$