1. **Problem statement:** Find the following: i) The perpendicular bisector of line segment $XY$. ii) The bisector of angle $YXZ$. iii) The perpendicular from point $Z$ to line $XY$. 2. **Formulas and rules:** - The **perpendicular bisector** of a segment is a line that is perpendicular to the segment and passes through its midpoint. - The **angle bisector** divides an angle into two equal angles. - The **perpendicular from a point to a line** is the shortest distance from the point to the line, forming a right angle. 3. **Step i: Perpendicular bisector of $XY$** - Find midpoint $M$ of $XY$: $$M = \left(\frac{x_X + x_Y}{2}, \frac{y_X + y_Y}{2}\right)$$ - Find slope of $XY$: $$m_{XY} = \frac{y_Y - y_X}{x_Y - x_X}$$ - Slope of perpendicular bisector: $$m_{\perp} = -\frac{1}{m_{XY}}$$ (negative reciprocal) - Equation of perpendicular bisector: $$y - y_M = m_{\perp}(x - x_M)$$ 4. **Step ii: Bisector of angle $YXZ$** - Angle $YXZ$ is formed at point $X$ by points $Y$ and $Z$. - To find the angle bisector, find unit vectors from $X$ to $Y$ and $X$ to $Z$: $$\vec{u} = \frac{\overrightarrow{XY}}{|\overrightarrow{XY}|}, \quad \vec{v} = \frac{\overrightarrow{XZ}}{|\overrightarrow{XZ}|}$$ - The angle bisector direction vector is: $$\vec{w} = \vec{u} + \vec{v}$$ - Equation of angle bisector line passing through $X$: $$\vec{r}(t) = \vec{X} + t\vec{w}$$ 5. **Step iii: Perpendicular from $Z$ to line $XY$** - Equation of line $XY$: $$y - y_X = m_{XY}(x - x_X)$$ - Slope of perpendicular from $Z$ to $XY$ is $m_{\perp} = -\frac{1}{m_{XY}}$ - Equation of perpendicular line from $Z$: $$y - y_Z = m_{\perp}(x - x_Z)$$ - Find intersection point $P$ of these two lines by solving the system: $$\begin{cases} y - y_X = m_{XY}(x - x_X) \\ y - y_Z = m_{\perp}(x - x_Z) \end{cases}$$ - Point $P$ is the foot of the perpendicular from $Z$ to $XY$. **Summary:** - The perpendicular bisector passes through midpoint of $XY$ with slope negative reciprocal of $XY$. - The angle bisector is along the vector sum of unit vectors from $X$ to $Y$ and $X$ to $Z$. - The perpendicular from $Z$ to $XY$ is found by intersecting the perpendicular line from $Z$ with $XY$. This completes the solution for the first problem (i).