Right Triangle Sides 495986

1. **Problem 1:** Given a right-angled triangle with one leg $3cm$ and hypotenuse $10\sqrt{2}cm$, find the other leg. 2. **Formula:** Use the Pythagorean theorem: $$c^2 = a^2 + b^2$$ where $c$ is the hypotenuse, and $a$, $b$ are the legs. 3. **Step 1:** Let the unknown side be $x$. Then: $$ (10\sqrt{2})^2 = 3^2 + x^2 $$ 4. **Step 2:** Calculate squares: $$ 100 \times 2 = 9 + x^2 $$ $$ 200 = 9 + x^2 $$ 5. **Step 3:** Solve for $x^2$: $$ x^2 = 200 - 9 = 191 $$ 6. **Step 4:** Write $x$ in surd form: $$ x = \sqrt{191} $$ 7. **Simplification:** $191$ is prime, so $\sqrt{191}$ is already simplified. --- 8. **Problem 2:** Given two sides $3\sqrt{3} cm$ and $5\sqrt{3} cm$ of a right triangle, find the remaining side. 9. **Step 1:** Identify which sides are legs or hypotenuse. Since $5\sqrt{3} > 3\sqrt{3}$, assume $5\sqrt{3}$ is hypotenuse. 10. **Step 2:** Let the unknown side be $y$. Apply Pythagorean theorem: $$ (5\sqrt{3})^2 = (3\sqrt{3})^2 + y^2 $$ 11. **Step 3:** Calculate squares: $$ 25 \times 3 = 9 \times 3 + y^2 $$ $$ 75 = 27 + y^2 $$ 12. **Step 4:** Solve for $y^2$: $$ y^2 = 75 - 27 = 48 $$ 13. **Step 5:** Simplify $y$: $$ y = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} $$ --- **Final answers:** - First triangle remaining side: $\sqrt{191} cm$ - Second triangle remaining side: $4\sqrt{3} cm$