1. **Problem Statement:** We have three points: $A(0,k)$, $B(6,-1)$, and $C(0,-3)$ in the coordinate plane. $L_1$ and $L_2$ are perpendicular bisectors of segments $AB$ and $BC$ respectively. Given: Equation of $L_1$ is $x - y - 1 = 0$. We need to: (a) Find the equation of $L_2$. (b) Find the coordinates of the intersection point of $L_1$ and $L_2$. (c) Prove the three perpendicular bisectors of triangle $ABC$ are concurrent. --- 2. **Recall:** - The perpendicular bisector of a segment passes through the midpoint and is perpendicular to the segment. - The three perpendicular bisectors of a triangle intersect at the circumcenter. --- 3. **(a) Find equation of $L_2$ (perpendicular bisector of $BC$):** - Points $B(6,-1)$ and $C(0,-3)$. - Midpoint $M_{BC} = \left(\frac{6+0}{2}, \frac{-1 + (-3)}{2}\right) = (3, -2)$. - Slope of $BC$ is $m_{BC} = \frac{-3 - (-1)}{0 - 6} = \frac{-2}{-6} = \frac{1}{3}$. - Slope of perpendicular bisector $L_2$ is negative reciprocal: $m_{L_2} = -3$. - Equation of $L_2$ using point-slope form: $$y - (-2) = -3(x - 3)$$ $$y + 2 = -3x + 9$$ $$y = -3x + 7$$ --- 4. **(b) Find intersection of $L_1$ and $L_2$:** - $L_1: x - y - 1 = 0 \Rightarrow y = x - 1$ - $L_2: y = -3x + 7$ - Set equal: $$x - 1 = -3x + 7$$ $$x + 3x = 7 + 1$$ $$4x = 8$$ $$x = 2$$ - Substitute $x=2$ into $y = x - 1$: $$y = 2 - 1 = 1$$ - Intersection point is $(2,1)$. --- 5. **(c) Prove concurrency of three perpendicular bisectors:** - The third perpendicular bisector is of segment $AC$. - Points $A(0,k)$ and $C(0,-3)$. - Midpoint $M_{AC} = \left(0, \frac{k + (-3)}{2}\right) = (0, \frac{k-3}{2})$. - Slope of $AC$ is $m_{AC} = \frac{-3 - k}{0 - 0}$ which is undefined (vertical line). - So $AC$ is vertical line $x=0$. - Perpendicular bisector of $AC$ is horizontal line through midpoint: $$y = \frac{k-3}{2}$$ - Check if this line passes through intersection point $(2,1)$: $$1 \stackrel{?}{=} \frac{k-3}{2} \Rightarrow 2 = k - 3 \Rightarrow k = 5$$ - Since $k=5$, point $A$ is $(0,5)$. - The three perpendicular bisectors intersect at $(2,1)$, confirming concurrency. --- **Final answers:** (a) $L_2: y = -3x + 7$ (b) Intersection point: $(2,1)$ (c) The three perpendicular bisectors meet at $(2,1)$ when $k=5$, proving concurrency.