Question

1. figure lmno is similar to figure tuvw. find the lengths of $overline{lm}$ and $overline{no}$

Explanation:

Step1: Set up proportion for similar - figures

Since figure $LMNO$ is similar to figure $TUVW$, the ratios of corresponding sides are equal. Let's assume that $\overline{LM}$ corresponds to $\overline{UV}$ and $\overline{NO}$ corresponds to $\overline{VW}$. Then $\frac{LM}{UV}=\frac{NO}{VW}$. We have $LM = 3x + 3$, $NO=6 - x$, $UV = 5$, and $VW = 3$. So, $\frac{3x + 3}{5}=\frac{6 - x}{3}$.

Step2: Cross - multiply

Cross - multiplying the proportion $\frac{3x + 3}{5}=\frac{6 - x}{3}$ gives us $3(3x + 3)=5(6 - x)$.
Expand both sides: $9x+9 = 30-5x$.

Step3: Solve for $x$

Add $5x$ to both sides: $9x + 5x+9=30-5x + 5x$, which simplifies to $14x+9 = 30$.
Subtract 9 from both sides: $14x+9 - 9=30 - 9$, so $14x=21$.
Divide both sides by 14: $x=\frac{21}{14}=\frac{3}{2}$.

Step4: Find the length of $\overline{LM}$

Substitute $x = \frac{3}{2}$ into the expression for $LM$: $LM=3x + 3=3\times\frac{3}{2}+3=\frac{9}{2}+3=\frac{9 + 6}{2}=\frac{15}{2}=7.5$.

Step5: Find the length of $\overline{NO}$

Substitute $x=\frac{3}{2}$ into the expression for $NO$: $NO=6 - x=6-\frac{3}{2}=\frac{12 - 3}{2}=\frac{9}{2}=4.5$.

Answer:

$LM = 7.5$, $NO = 4.5$