Triangle Region 45F221

1. **Problem Statement:** We are given a triangular region in the coordinate plane with vertices at $$(2,4)$$, $$(2,5)$$, and $$(4,5)$$. We want to understand the shape and boundaries of this triangle. 2. **Identify the vertices:** The vertices are: - Point A: $$(2,4)$$ - Point B: $$(2,5)$$ - Point C: $$(4,5)$$ 3. **Find the equations of the sides:** - Side AB is vertical at $$x=2$$ between $$y=4$$ and $$y=5$$. - Side BC is horizontal at $$y=5$$ between $$x=2$$ and $$x=4$$. - Side AC connects $$(2,4)$$ to $$(4,5)$$. The slope is $$m=\frac{5-4}{4-2}=\frac{1}{2}$$. Equation of line AC: $$y - 4 = \frac{1}{2}(x - 2)$$ $$y = \frac{1}{2}x + 3$$ 4. **Summary of boundaries:** - Left boundary: $$x=2$$ for $$4 \leq y \leq 5$$ - Top boundary: $$y=5$$ for $$2 \leq x \leq 4$$ - Hypotenuse boundary: $$y=\frac{1}{2}x + 3$$ for $$2 \leq x \leq 4$$ 5. **Interpretation:** The shaded triangular region is bounded by these three lines and lies in the top-right quadrant. **Final answer:** The triangle has vertices at $$(2,4)$$, $$(2,5)$$, and $$(4,5)$$ with boundaries: $$x=2$$, $$y=5$$, and $$y=\frac{1}{2}x + 3$$.