1. **State the problem:** We are given a right triangle $PQR$ with right angle at $R$. A perpendicular segment $TR$ is drawn from $R$ to $PQ$, forming two right angles at $T$ and $R$. We know $PS=8$ units, $QR=21$ units, and $PQ=\sqrt{865}$ units. We need to find the area of triangle $PST$. 2. **Understand the figure and given data:** Since $PQR$ is a right triangle with right angle at $R$, by the Pythagorean theorem: $$PQ^2 = PR^2 + QR^2$$ We know $PQ=\sqrt{865}$, so: $$865 = PR^2 + 21^2$$ $$865 = PR^2 + 441$$ $$PR^2 = 865 - 441 = 424$$ $$PR = \sqrt{424}$$ 3. **Locate point $S$ and segment $PS$:** The problem states $PS=8$ units. Since $S$ lies on $PR$ (implied by the figure and naming), $S$ divides $PR$ such that $PS=8$. 4. **Find coordinates for points (conceptual):** To find the area of triangle $PST$, we can use coordinate geometry. - Place $R$ at origin $(0,0)$. - Since $QR=21$ and $R$ is right angle, place $Q$ at $(0,21)$. - Place $P$ at $(x,0)$ where $x=PR=\sqrt{424}$. 5. **Coordinates of points:** - $P = (\sqrt{424},0)$ - $Q = (0,21)$ - $R = (0,0)$ 6. **Find point $S$ on $PR$ such that $PS=8$:** Vector $PR = P - R = (\sqrt{424},0)$ Since $S$ lies on $PR$ starting from $P$, moving towards $R$, the vector $PS$ has length 8. So, $$S = P - 8 \cdot \frac{PR}{|PR|} = (\sqrt{424},0) - 8 \cdot \frac{(\sqrt{424},0)}{\sqrt{424}} = (\sqrt{424} - 8, 0)$$ 7. **Find point $T$ on $PQ$ such that $TR$ is perpendicular to $PQ$:** Vector $PQ = Q - P = (0 - \sqrt{424}, 21 - 0) = (-\sqrt{424}, 21)$ Parametric form of $PQ$: $$T = P + t \cdot PQ = (\sqrt{424},0) + t(-\sqrt{424},21) = (\sqrt{424} - t\sqrt{424}, 21t)$$ 8. **Since $TR$ is perpendicular to $PQ$, vector $TR$ is perpendicular to $PQ$:** Vector $TR = R - T = (0 - (\sqrt{424} - t\sqrt{424}), 0 - 21t) = (-\sqrt{424} + t\sqrt{424}, -21t)$ Dot product $TR \cdot PQ = 0$: $$(-\sqrt{424} + t\sqrt{424})(-\sqrt{424}) + (-21t)(21) = 0$$ $$\sqrt{424} \cdot \sqrt{424} - t \sqrt{424} \cdot \sqrt{424} - 441t = 0$$ $$424 - 424t - 441t = 0$$ $$424 - 865t = 0$$ $$t = \frac{424}{865}$$ 9. **Coordinates of $T$:** $$x_T = \sqrt{424} - \frac{424}{865} \cdot \sqrt{424} = \sqrt{424} \left(1 - \frac{424}{865}\right) = \sqrt{424} \cdot \frac{865 - 424}{865} = \sqrt{424} \cdot \frac{441}{865}$$ $$y_T = 21 \cdot \frac{424}{865} = \frac{8904}{865}$$ 10. **Coordinates of $P$, $S$, and $T$:** - $P = (\sqrt{424}, 0)$ - $S = (\sqrt{424} - 8, 0)$ - $T = \left(\sqrt{424} \cdot \frac{441}{865}, \frac{8904}{865}\right)$ 11. **Calculate area of triangle $PST$ using coordinate formula:** $$\text{Area} = \frac{1}{2} \left| x_P(y_S - y_T) + x_S(y_T - y_P) + x_T(y_P - y_S) \right|$$ Since $y_P = y_S = 0$, this simplifies to: $$\text{Area} = \frac{1}{2} |x_P(0 - y_T) + x_S(y_T - 0) + x_T(0 - 0)| = \frac{1}{2} | -x_P y_T + x_S y_T | = \frac{1}{2} |y_T (x_S - x_P)|$$ 12. **Calculate $x_S - x_P$:** $$x_S - x_P = (\sqrt{424} - 8) - \sqrt{424} = -8$$ 13. **Calculate area:** $$\text{Area} = \frac{1}{2} |y_T \cdot (-8)| = \frac{1}{2} \cdot 8 \cdot y_T = 4 y_T$$ Substitute $y_T$: $$4 \cdot \frac{8904}{865} = \frac{35616}{865}$$ 14. **Final answer:** The area of triangle $PST$ is $$\boxed{\frac{35616}{865} \text{ square units}}$$ This is approximately $41.16$ square units.