1. **Problem Statement:** Find the length of the missing side in a right triangle using the Pythagorean theorem, given sides $a$, $b$, and hypotenuse $c$. 2. **Formula:** The Pythagorean theorem states: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse (longest side), and $a$, $b$ are the other two sides. 3. **Example Calculation:** Given $a^2 + b^2 = 8^2$, and one side $b = 6$, substitute: $$a^2 + 6^2 = 8^2$$ $$a^2 + 36 = 64$$ 4. **Solve for $a^2$:** $$a^2 = 64 - 36$$ $$a^2 = 28$$ 5. **Find $a$ by taking the square root:** $$a = \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \approx 5.29$$ 6. **Triangle Inequality:** For any triangle with sides $x$, $y$, and $z$, the sum of any two sides must be greater than the third: $$x + y > z$$ $$x + z > y$$ $$y + z > x$$ 7. **Given sides 9 and 6, find possible $x$ for the third side:** $$9 + 6 > x \Rightarrow 15 > x$$ $$9 - 6 < x \Rightarrow 3 < x$$ So the possible side lengths satisfy: $$3 < x < 15$$ 8. **Distance between points $(6,7)$ and $(-5,-3)$:** Use the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Substitute: $$d = \sqrt{(-5 - 6)^2 + (-3 - 7)^2} = \sqrt{(-11)^2 + (-10)^2} = \sqrt{121 + 100} = \sqrt{221} \approx 14.87$$ **Final answers:** - Missing side length $a \approx 5.29$ - Possible third side lengths $x$ satisfy $3 < x < 15$ - Distance between points is approximately $14.87$ units.