Question

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

Explanation:

Step1: Use congruent - segment property

Since $\overline{LK}\cong\overline{MK}$, then $LK = MK$. Also, $MK=MN - KN$. So we set up the equation $7x - 10=(9x - 11)-(x + 3)$.

Step2: Simplify the right - hand side of the equation

Simplify $(9x - 11)-(x + 3)$:
\[

$$\begin{align*} (9x - 11)-(x + 3)&=9x-11 - x - 3\\ &=(9x - x)+(-11 - 3)\\ &=8x-14 \end{align*}$$

\]
So the equation becomes $7x - 10 = 8x-14$.

Step3: Solve for $x$

Subtract $7x$ from both sides: $- 10=x - 14$.
Add 14 to both sides: $x = 4$.

Step4: Find $LK$

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

Step5: Calculate $LJ$

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

Answer:

$46$