Question

1. given $\triangle ghi \sim \triangle wvu$, what is the length of $\overline{gi}$?

1. given $\triangle def \sim \triangle uvw$, what is the length of $\overline{uw}$?

Explanation:

Step1: Find similarity ratio (Q3)

Identify corresponding sides: $\overline{GH}$ corresponds to $\overline{WU}$. Calculate ratio:
$\text{Ratio} = \frac{GH}{WU} = \frac{14.4}{6} = 2.4$

Step2: Find $\overline{GI}$ (Q3)

$\overline{GI}$ corresponds to $\overline{WU}$'s hypotenuse $\overline{WU}$? No, $\overline{GI}$ corresponds to $\overline{UW}$. Use ratio to find $\overline{GI}$:
$\overline{GI} = \overline{UW} \times 2.4 = 6 \times 2.4 = 14.4$? Correction: $\overline{GH}$ (14.4) corresponds to $\overline{WU}$ (6), $\overline{GI}$ corresponds to $\overline{UW}$. First confirm $\overline{UW}$ is hypotenuse of $\triangle WVU$, length 6. So:
$\overline{GI} = 6 \times 2.4 = 14.4$? No, correction: $\overline{GH}$ corresponds to $\overline{WV}$ (4.8). Ratio = $\frac{14.4}{4.8}=3$. Then $\overline{GI}$ corresponds to $\overline{WU}$ (6):
$\overline{GI} = 6 \times 3 = 18$

Step3: Find similarity ratio (Q4)

$\overline{DE}$ (12) corresponds to $\overline{UV}$. Ratio = $\frac{UV}{DE} = \frac{UV}{12}$. $\overline{EF}$ (5) corresponds to $\overline{VW}$ (2), so ratio = $\frac{2}{5}=0.4$

Step4: Find $\overline{UW}$ (Q4)

First find hypotenuse $\overline{DF}$ of $\triangle DEF$:
$\overline{DF} = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169}=13$
Then $\overline{UW} = \overline{DF} \times 0.4 = 13 \times 0.4 = 5.2$

Answer:

1. The length of $\overline{GI}$ is 18
2. The length of $\overline{UW}$ is 5.2