1. **State the problem:** We have two right triangles, $\triangle ACR$ and $\triangle YZX$. $\triangle ACR$ has sides $AC=16$, $CR=30$, and hypotenuse $AR=34$. $\triangle YZX$ has sides $YZ=15$, $ZX=r$, and hypotenuse $YX=p$. We want to find the values of $r$ and $p$ assuming the triangles are similar. 2. **Formula and rules:** For right triangles, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. 3. **Check the first triangle:** Verify $AR$ is the hypotenuse: $$16^2 + 30^2 = 256 + 900 = 1156$$ $$34^2 = 1156$$ Since both sides equal, the triangle is right-angled at $C$. 4. **Find the scale factor between triangles:** Since $YZ=15$ corresponds to $AC=16$, the scale factor from $\triangle ACR$ to $\triangle YZX$ is: $$k = \frac{15}{16}$$ 5. **Find $r$:** Side $ZX$ corresponds to $CR=30$, so: $$r = 30 \times k = 30 \times \frac{15}{16} = \frac{450}{16} = 28.125$$ 6. **Find $p$:** Hypotenuse $YX$ corresponds to $AR=34$, so: $$p = 34 \times k = 34 \times \frac{15}{16} = \frac{510}{16} = 31.875$$ 7. **Summary:** - $r = 28.125$ - $p = 31.875$ These values maintain similarity between the two right triangles.