Question

1. if $overline{pr}congoverline{qs}$, what are the lengths of $overline{pq}$, $overline{qr}$, and $overline{rs}$? (5x + 3)-(7x - 14)=0 - 2x+17 = 0 - 2x=-17 x = 8.5

Explanation:

Step1: Set up the equation based on congruence

Since $\overline{PR}\cong\overline{QS}$, we know that the lengths of $PR$ and $QS$ are equal. The length of $PR=(3x - 3)+(4x-11)=7x-14$, and the length of $QS=(2x + 6)+(3x-3)=5x + 3$. So, we set up the equation $7x-14=5x + 3$.

Step2: Solve the equation for x

Subtract $5x$ from both sides: $7x-5x-14=5x-5x + 3$, which simplifies to $2x-14=3$. Then add 14 to both sides: $2x-14 + 14=3+14$, giving $2x=17$. Divide both sides by 2: $x=\frac{17}{2}=8.5$.

Step3: Find the length of $\overline{PQ}$

Substitute $x = 8.5$ into the expression for the length of $\overline{PQ}$, which is $2x+6$. So, $PQ=2\times8.5 + 6=17 + 6=23$.

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

Substitute $x = 8.5$ into the expression for the length of $\overline{QR}$, which is $3x-3$. So, $QR=3\times8.5-3=25.5 - 3=22.5$.

Step5: Find the length of $\overline{RS}$

Substitute $x = 8.5$ into the expression for the length of $\overline{RS}$, which is $4x-11$. So, $RS=4\times8.5-11=34 - 11=23$.

Answer:

$PQ = 23$, $QR=22.5$, $RS = 23$