Question

1. solve for the missing lengths. show all work.

a. △ghi ~ △pqr
b. △abc ~ △xyz

Explanation:

Step1: Recall property of similar triangles

For similar triangles $\triangle GHI\sim\triangle PQR$ and $\triangle ABC\sim\triangle XYZ$, the ratios of corresponding - sides are equal. Let's assume for $\triangle GHI\sim\triangle PQR$, if $GH = a$, $HI = b$, $GI = c$, $PQ = d$, $QR = e$, $PR = f$, then $\frac{a}{d}=\frac{b}{e}=\frac{c}{f}$. Similarly for $\triangle ABC\sim\triangle XYZ$, if $AB = m$, $BC = n$, $AC = p$, $XY = q$, $YZ = r$, $XZ = s$, then $\frac{m}{q}=\frac{n}{r}=\frac{p}{s}$. But we need to know which sides are corresponding. Since the figure has no side - length labels for $\triangle GHI$ and $\triangle PQR$ other than one side of $\triangle PQR$ is 112, and for $\triangle ABC$, $AB = 7$, $AC = 23$ and for $\triangle XYZ$, $XZ = 49$. Let's assume the corresponding - side ratios for $\triangle ABC\sim\triangle XYZ$: $\frac{AB}{XY}=\frac{AC}{XZ}$.

Step2: Set up proportion for $\triangle ABC\sim\triangle XYZ$

Let $XY = w$. We have $\frac{7}{w}=\frac{23}{49}$. Cross - multiply to get $23w=7\times49$.

Step3: Solve for $w$

$w=\frac{7\times49}{23}=\frac{343}{23}\approx14.91$. For $\triangle GHI\sim\triangle PQR$, since we have no information about which sides are corresponding and no other side - lengths for $\triangle GHI$ except the shape, we cannot solve for the missing lengths.

Answer:

For $\triangle ABC\sim\triangle XYZ$, if $XY = w$, then $w=\frac{343}{23}\approx14.91$. For $\triangle GHI\sim\triangle PQR$, not enough information to solve for missing lengths.