Shortest Distance Eb0747

1. **State the problem:** We want to find which point among $(2,-4)$, $(4,2)$, $(3,-1)$, and $(1,-7)$ on the line segment $DE$ is the shortest distance from point $F(-8,2)$. 2. **Find the equation of line $DE$:** - Points $D(6,6)$ and $E(5,2)$. - Slope $m = \frac{6-2}{6-5} = \frac{4}{1} = 4$. - Equation using point-slope form: $$y - 2 = 4(x - 5)$$ $$y = 4x - 20 + 2 = 4x - 18$$ 3. **Check which given points lie on line $DE$:** - For $(2,-4)$: $y = 4(2) - 18 = 8 - 18 = -10 \neq -4$ (not on line) - For $(4,2)$: $y = 4(4) - 18 = 16 - 18 = -2 \neq 2$ (not on line) - For $(3,-1)$: $y = 4(3) - 18 = 12 - 18 = -6 \neq -1$ (not on line) - For $(1,-7)$: $y = 4(1) - 18 = 4 - 18 = -14 \neq -7$ (not on line) None of these points lie exactly on the line $DE$. However, the problem states "Which point on $DE$ is the shortest distance away from $F$?" and lists these points, so we interpret these as candidate points to check distance from $F$. 4. **Calculate distances from $F(-8,2)$ to each candidate point:** - Distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ - To $(2,-4)$: $$d = \sqrt{(2 + 8)^2 + (-4 - 2)^2} = \sqrt{10^2 + (-6)^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66$$ - To $(4,2)$: $$d = \sqrt{(4 + 8)^2 + (2 - 2)^2} = \sqrt{12^2 + 0^2} = \sqrt{144} = 12$$ - To $(3,-1)$: $$d = \sqrt{(3 + 8)^2 + (-1 - 2)^2} = \sqrt{11^2 + (-3)^2} = \sqrt{121 + 9} = \sqrt{130} \approx 11.40$$ - To $(1,-7)$: $$d = \sqrt{(1 + 8)^2 + (-7 - 2)^2} = \sqrt{9^2 + (-9)^2} = \sqrt{81 + 81} = \sqrt{162} \approx 12.73$$ 5. **Compare distances:** - $(2,-4)$: $\approx 11.66$ - $(4,2)$: $12$ - $(3,-1)$: $\approx 11.40$ - $(1,-7)$: $\approx 12.73$ 6. **Conclusion:** The point $(3,-1)$ is the closest to $F$ among the given options. **Final answer:** $(3,-1)$