1. **State the problem:** We need to find the unknown side length in each right triangle using the Pythagorean theorem. 2. **Recall the Pythagorean theorem:** For a right triangle with legs $a$ and $b$, and hypotenuse $c$, the relationship is: $$c^2 = a^2 + b^2$$ 3. **Top-left triangle:** Given base $8$ and hypotenuse $10$, find the vertical leg $x$. Apply the formula: $$10^2 = 8^2 + x^2$$ Simplify: $$100 = 64 + x^2$$ Subtract 64 from both sides: $$100 - 64 = x^2$$ $$36 = x^2$$ Take the square root: $$x = \sqrt{36} = 6$$ 4. **Center triangle:** Given one leg $2$ and hypotenuse $5$, find the unknown bottom side $y$. Apply the formula: $$5^2 = 2^2 + y^2$$ Simplify: $$25 = 4 + y^2$$ Subtract 4 from both sides: $$25 - 4 = y^2$$ $$21 = y^2$$ Take the square root: $$y = \sqrt{21}$$ **Final answers:** - Top-left triangle unknown side: $6$ - Center triangle unknown side: $\sqrt{21}$