Question

triangle nbt is similar to triangle yld.

1. which segment corresponds with segment ld?
2. what is the constant of proportionality between triangle nbt and triangle yld?
3. what is the length of side yd?
4. what is the length of side ly?

Explanation:

Step1: Match corresponding sides

First, identify the side in $\triangle NBT$ that corresponds to $LD$ by matching the similar triangle vertices $\triangle NBT \sim \triangle YLD$. Vertex $L$ corresponds to $B$, vertex $D$ corresponds to $T$, so $LD$ corresponds to $BT$.

Step2: Find proportionality constant

Calculate the ratio of corresponding sides of $\triangle NBT$ to $\triangle YLD$. Use the known sides $BT=24$ and $LD=16$.
$\text{Constant} = \frac{BT}{LD} = \frac{24}{16} = \frac{3}{2}$

Step3: Calculate length of $YD$

$YD$ corresponds to $NT=27$. Use the proportionality constant: $\frac{NT}{YD} = \frac{3}{2}$, so $YD = NT \times \frac{2}{3}$
$YD = 27 \times \frac{2}{3} = 18$

Step4: Calculate length of $LY$

$LY$ corresponds to $NB=30$. Use the proportionality constant: $\frac{NB}{LY} = \frac{3}{2}$, so $LY = NB \times \frac{2}{3}$
$LY = 30 \times \frac{2}{3} = 20$

Answer:

1. Segment $BT$
2. $\frac{3}{2}$
3. 18
4. 20