Bisectors

In Geometry, Bisectors are defined as line segments that divide shapes, other line segements or angles into 2 equal parts. The 2 main types of bisectors are Perpendicular Bisectors and Angle Bisectors.

Perpendicular Bisectors

Perpendicular Bisectors are line segments that are perpendicular to another line segment and pass through the midpoint of that segment. Any point on the bisector is equidistant from the endpoints of both sides of the other line segment.

In the example shown above, the line segment CD acts as a perpendicular bisector to line segment AD as indicated by the right angle it forms. In addition, enpoints \(\text{A}\) and \(\text{B}\) are equidistant from \(\text{CD}\).

Using the original line segment, we can determine the perpendicular bisector equation using the following process:

1. Determine the midpoint of the line segment using the midpoint formula: \([\cfrac{x₁ + x₂}{2}, \cfrac{y₁ + y₂}{2}]\)
2. Determine the slope of the line segment using the slope formula: \(\cfrac{y₂ - y₁}{x₂ - x₁}\)
3. Determine the slope of the perpendicular bisector by taking the negative reciprocol of the original line segment's slope: \(\cfrac{-1}{m}\)
4. Plug the midpoint and the slope into the slope-point equation (\(y = mx + b\)) to determine the y-intecept for the perpendicular bisector
5. Put everything together to find the equation fo the perpendicular bisector

Example

Find the equation of a line perpendicular to the line \(2y + 8 = 4x\) that travels through \((6,5)\).

As we already have the midpoint and slope of the original equation, we can skip Steps 1 and 2.
We can determine the slope of the perpendicular bisector by rearranging the original equation and taking the negative reciprocol of its slope:

\(2y = 4x - 8\)
\(\cfrac{\cancel{2}y}{\cancel{2}} = \cfrac{4x}{2} - \cfrac{8}{2}\)
\(y = 2x - 4\)
m_{perpendicular} \(\cfrac{-1}{2}\)

Now that we have the slope of the perpendicular bisector, we can plug the values of the midpoint into the slope-point eqution to determine the y-intercept of the perpendicular bisector equation:

\(5 = (\cfrac{-1}{2})(6) + b\)
\(5 = -3 + b\)
\(b = 5 + 3 = 8\)

Therefore, we can determine that the equation for the perpendicular bisector is \(y = \cfrac{-1}{2} + 8\)

---

Angle Bisectors

Angle Bisectors are line segments that divide angles into 2 congruent (or equal) angles. Any point on the angle bisector is equidistant from both sides of the angle.

In the example shown above, the angle bisector divides the \(90\)° angle into 2 seperate \(45\)° angles.

As a triangle contains 3 vertices and 3 angles, this always results in 3 angle bisectors. The angle bisectors intersect at a common point called the incenter which is equidistant from all the vertices. An Angle Bisector incorporates the Angle Bisector Theorem, stating that when the line segment bisects an angle, the ratio of the 2 line segments is equal to that of the other 2 sides. It is only used when most of the side lengths and the angle bisector is known:

In the example shown above, we can identify that \(\cfrac{\text{AD}}{\text{CD}} = \cfrac{\text{BA}}{\text{BC}}\).

Example

Determine the value of \(x\) for △\(\text{PQR}\) using the bisector angle theorem.

Using the Angle Bisector Theorem, we can identify the ratios as \(\cfrac{\text{RD}}{\text{QD}}\ = \cfrac{\text{PR}}{\text{PQ}}\).
Inserting values into these fractions gives us \(\cfrac{x}{20}\ = \cfrac{15}{30}\).

We can cross-multiply in order to determine \(x\):

\((\cancel{20})(\cfrac{x}{\cancel{20}})\ = (20)(\cfrac{15}{30})\)

\(x = \cfrac{(15)(20)}{30}\)

\(x = \cfrac{300}{30}\)

\(x = 10\)

Therefore, we can determine that \(x = 10\).

---