1. **Problem 1:** Find $x$ and $y$ in the first figure. Given: Triangle $ABC$ with right angle at $D$, $AD=3\sqrt{2}$, $AC=5$, $AB=6$, $CD=x$, $DB=y$. 2. **Step 1: Understand the triangle and use Pythagoras theorem.** Since $AD$ is the height from $A$ to $CB$, and $D$ is right angle, triangles $ADC$ and $ADB$ are right triangles. 3. **Step 2: Use Pythagoras theorem in triangle $ADC$.** $$AC^2 = AD^2 + CD^2$$ Substitute known values: $$5^2 = (3\sqrt{2})^2 + x^2$$ $$25 = 18 + x^2$$ $$x^2 = 25 - 18 = 7$$ $$x = \sqrt{7}$$ 4. **Step 3: Use Pythagoras theorem in triangle $ADB$.** $$AB^2 = AD^2 + DB^2$$ Substitute known values: $$6^2 = (3\sqrt{2})^2 + y^2$$ $$36 = 18 + y^2$$ $$y^2 = 36 - 18 = 18$$ $$y = \sqrt{18} = 3\sqrt{2}$$ 5. **Final answers for problem 1:** $$x = \sqrt{7}$$ $$y = 3\sqrt{2}$$