Question

1. in the diagram, △pqr is the image of △pqr under a dilation with center o and scale factor $\frac{2}{3}$. find the length of $overline{pq}$.

Explanation:

Step1: Use dilation - length relationship

The ratio of the lengths of corresponding sides of a dilated figure and the original figure is equal to the scale - factor. For corresponding sides $\overline{PQ}$ and $\overline{P'Q'}$, we have the equation $\frac{P'Q'}{PQ}=\frac{2}{3}$. Given $P'Q'=\frac{8}{3}$ and $PQ=x + 1$. So, $\frac{\frac{8}{3}}{x + 1}=\frac{2}{3}$.

Step2: Cross - multiply

Cross - multiplying the equation $\frac{\frac{8}{3}}{x + 1}=\frac{2}{3}$ gives us $2(x + 1)=\frac{8}{3}\times3$.

Step3: Simplify the right - hand side

$\frac{8}{3}\times3 = 8$, so the equation becomes $2(x + 1)=8$.

Step4: Distribute the 2

Using the distributive property $a(b + c)=ab+ac$, we get $2x+2 = 8$.

Step5: Solve for x

Subtract 2 from both sides: $2x=8 - 2$, so $2x=6$. Then divide both sides by 2: $x = 3$.

Step6: Find the length of $\overline{PQ}$

Since $PQ=x + 1$ and $x = 3$, then $PQ=3 + 1=4$.

Answer:

4