Question

1. given $\triangle mno \sim \triangle pqr$, find the lengths of $\overline{no}$, $\overline{pq}$, and $\overline{qr}$.

Explanation:

Step1: Find similarity ratio

$\text{Ratio} = \frac{PR}{OM} = \frac{82}{41} = 2$

Step2: Calculate $PQ$ (corresponds to $MN$)

$PQ = MN \times \text{Ratio} = 9 \times 2 = 18$

Step3: Find $NO$ via Pythagoras

$NO = \sqrt{OM^2 - MN^2} = \sqrt{41^2 - 9^2} = \sqrt{1681 - 81} = \sqrt{1600} = 40$

Step4: Calculate $QR$ (corresponds to $NO$)

$QR = NO \times \text{Ratio} = 40 \times 2 = 80$

Answer:

$NO = 40$, $PQ = 18$, $QR = 80$