1. The problem asks to find the value of $h$ for point $Q(h, 5)$, given points $P(-7, -3)$ and $R(8, 9)$ forming a right triangle with $Q$ on the hypotenuse. 2. Since $Q$ lies on the line segment $PR$, the coordinates of $Q$ must satisfy the equation of the line passing through $P$ and $R$. 3. Find the slope $m$ of line $PR$ using points $P$ and $R$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - (-3)}{8 - (-7)} = \frac{12}{15} = \frac{4}{5}$$ 4. Using point-slope form with point $P(-7, -3)$, the equation of line $PR$ is: $$y - (-3) = \frac{4}{5}(x - (-7))$$ $$y + 3 = \frac{4}{5}(x + 7)$$ 5. Substitute $y = 5$ (the $y$-coordinate of $Q$) to solve for $x = h$: $$5 + 3 = \frac{4}{5}(h + 7)$$ $$8 = \frac{4}{5}(h + 7)$$ 6. Multiply both sides by 5: $$40 = 4(h + 7)$$ 7. Divide both sides by 4: $$10 = h + 7$$ 8. Solve for $h$: $$h = 10 - 7 = 3$$ Final answer: $\boxed{3}$