1. **State the problem:** We have two similar triangles, a large right triangle VZX and a smaller triangle WYX inside it. Given lengths VW = 48, WX = 32, and YX = 30, we need to find the length YZ. 2. **Identify the similarity and ratios:** Since triangles VZX and WYX are similar, corresponding sides are proportional. The sides VW and WX correspond to WY and YX respectively. 3. **Set up the ratio:** The ratio of sides in the smaller triangle to the larger triangle is $ \frac{WY}{VW} = \frac{YX}{WX} $. 4. **Calculate WY:** Using the ratio, $$\frac{WY}{48} = \frac{30}{32}$$ Multiply both sides by 48: $$WY = 48 \times \frac{30}{32}$$ Simplify: $$WY = \cancel{48} \times \frac{30}{\cancel{32}} \times \frac{1}{\cancel{1.5}} = 45$$ 5. **Find YZ:** Since YZ is the segment from Y to Z along the base, and WX = 32, YX = 30, the length YZ is the difference between WX and YX: $$YZ = WX - YX = 32 - 30 = 2$$ 6. **Final answer:** The length of YZ is $2$.