1. Stating the problem: We need to find the value of $h$ for the right triangle with vertices $P(-7,-3)$, $Q(h,5)$, and $R(8,9)$, where the right angle is at point $P$. 2. Since the right angle is at $P$, the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ are perpendicular. 3. Calculate vectors: $$\overrightarrow{PQ} = (h - (-7), 5 - (-3)) = (h + 7, 8)$$ $$\overrightarrow{PR} = (8 - (-7), 9 - (-3)) = (15, 12)$$ 4. For vectors to be perpendicular, their dot product equals zero: $$\overrightarrow{PQ} \cdot \overrightarrow{PR} = 0$$ $$ (h+7)(15) + (8)(12) = 0 $$ $$ 15h + 105 + 96 = 0 $$ $$ 15h + 201 = 0 $$ 5. Solve for $h$: $$ 15h = -201 $$ $$ h = \frac{-201}{15} = -13.4 $$ Final answer: $h = -13.4$