1. **State the problem:** We have a rectangle with height 12 in and base 8 in. Inside it, a right triangle is shaded with height $x$ in and base 8 in. The area of the shaded triangle is 78 in$^2$. We need to find $x$. 2. **Formula for area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Apply the formula:** Given base = 8 in and height = $x$ in, $$78 = \frac{1}{2} \times 8 \times x$$ 4. **Simplify the equation:** $$78 = 4x$$ 5. **Solve for $x$:** Divide both sides by 4: $$\frac{78}{\cancel{4}} = \frac{4x}{\cancel{4}}$$ $$x = \frac{78}{4}$$ 6. **Calculate the value:** $$x = 19.5$$ **Final answer:** $$x = 19.5 \text{ in}$$