**The perpendicular bisector theorem states that if a point lies on the perpendicular bisector of a line segment, it will be at an equal distance/equidistant from both endpoints of that line segment.**

## What Is Perpendicular Bisector Theorem?

The perpendicular bisector theorem is a theorem stating that if we take any point on the perpendicular bisector of a line segment,** then that point will be equidistant from both the endpoints of the line segment**. This is shown in the figure below.

*According to perpendicular bisector theorem:*

$CA = CB$

$DA = DB$

$EA = EB$

### Perpendicular Bisector

Consider two line segments, “$AB$” and “$CD$”. If the two segments cut each other in a manner that an angle of $90^{o}$ is formed, **then they are perpendicular to each other**.

If the line segment “$AB$” cuts the line segment “$CD$” such that it divides the line segment “$CD$” into two equal parts, then we will say that both these lines bisect each other. So if the line segment “$AB$” bisects line segment “$CD$” at a $90^{o}$ angle,** it will give us the perpendicular bisector**.

**Note**: In the above example, we can take a line or a ray instead of line segment “$AB$” as long as it is still bisecting the line segment “$CD$” at a $90^{o}$ angle. But we cannot take a line/ray instead of the line segment “$CD$” as a line/ray has an infinite length and cannot be cut in two equal halves.

## How To Use Perpendicular Bisector Theorem

We can use the perpendicular bisector theorem to **determine the missing lengths of the sides of a triangle** if sufficient data regarding the triangle is already given. Perpendicular bisector theorem can also be used along with other theorems to solve for lengths of a triangle.

Consider an example of a weather monitoring tower that is erected at a $90^{o}$ angle in the center of a piece of land. The land is $800$m in length while the tower’s height is $250$ meters, and we want to attach two guy-wires from the top of the tower to the end of the ground. **Perpendicular bisector theorem and Pythagoras theorem** will help us determine the length of the guy-wires.

The tower is like a perpendicular bisector for the land, so** it bisects the land into two equal parts of** $400$ **meters**. The height of the tower is given as 250 meter, so let us calculate the length of one guy-wire using Pythagoras theorem.

$c^{2}= 400^{2} + 250^{2}$

$c^{2} = 160,000 + 62,500$

$c^{2} = 222,500$

$c = \sqrt{222,500} = 472$ meter approx.

We know that any point on the perpendicular bisector is** at equal distance from both ends**, so the length of the other guy wire is also $472$ meter approx.

We used the perpendicular bisector theorem to** calculate the missing length of the triangle sides** in the above example. The conditions for utilization of perpendicular bisector are simple and* can be stated as:*

- The line, ray or line segment must bisect the other line segment at a $90^{o}$ angle.
- We must have sufficient data regarding the problem to solve for the remaining sides of the triangle.

### Proof of Perpendicular Bisector Theorem

It is a pretty straightforward proof. Let us draw a bisector on the line segment XY. **The spot where the bisector touches the line segment is M**, and we have to prove that the lines drawn from point C on the bisector to the endpoints X and Y are congruent or equal to each other.

If we assume that the line CM is a perpendicular bisector of the line segment XY, then this means** it bisects the XY at a** $90^{0}$** angle** and that the point M is the middle point of the line segment XY. Then by the definition of a perpendicular bisector, we have divided the line segment into two equal parts, so XM and MY are congruent.

$XM = MY$

If we draw two lines from the point $C$ to the endpoints of line segment $X$ and $Y$, we will get** two right-angle triangles** $XMC$ **and** $YMC$. We have already concluded that the XM and MY are congruent. Similarly, the bisector length for both triangles will also be the same.

$CM = CM$ ( for both triangles)

We have established that **two sides and one angle** (the $90^{0}$ one) **of the two triangles** $XMC$ **and** $YMC$ **are equal**. So by SAS congruent criteria, we know that angles $XMC$ and $YMC$ are congruent.

This provides us with the conclusion that the sides $CX$ and $CY$ **are congruent**.

### Proof of Converse Perpendicular Bisector Theorem

The converse perpendicular bisector theorem reverses the hypothesis of the original theorem. It states that **if point M is equidistant from both endpoints of the line segment** $XY$,** it is a perpendicular bisector of that line segment**.

Using the same picture above, if $CX = CY$,

Then we have to prove that $XM = YM$.

Draw a perpendicular line from point $C$ such that it cuts the line segment at point M.

Now compare $\triangle XMC$ and $\triangle YMC$:

$CX = CY$

$CM = CM$ ( for both traingles)

$\angle XMC = \angle YMC = 90^{o}$

So $\triangle XMC \cong \triangle YMC$ by SAS congruent criteria. Hence, $XM = YM$ **is proved**.

### Applications of Perpendicular Bisector Theorem

*some of which include:*

1. It is extensively used in the construction of bridges.

2. It is also used for the erection of towers and installing guy-wires around it.

3. It is used in making tables of different sizes and lengths.

### Example 1:

For the figure given below, calculate the value of “$x$”.

__Solution:__

We know that for a perpendicular bisector, the side $AC = BC$.

$6x\hspace{1mm} +\hspace{1mm}12 = 24$

$6x = 24\hspace{1mm} -\hspace{1mm}12$

$6x = 12$

$x = \dfrac{12}{6} = 2$

### Example 2:

Solve the unknown values of the triangle by using properties of the perpendicular bisector theorem.

__Solution:__

We know that the angle where perpendicular bisector bisects is equal to $90^{o}$.

$4x\hspace{1mm} + \hspace{1mm}10 = 90$

$4x = 80$

$x = 40^{o}$

The perpendicular bisector will divide the given length of $40 cm$ into two equal parts of $20 cm$ each. Hence, $2y – 4$ **will be equal to** $20 cm$.

$2y – 4 = 20$

$2y = 24$

$y = 12 cm$

### Example 3:

Using the properties of the perpendicular bisector theorem, calculate the value of “x” for the figure given below.

__Solution:__

From the properties of the perpendicular bisector theorem, **we know that the side **$AB = BC$.

$6x\hspace{1mm} +\hspace{1mm}4 = 8x\hspace{1mm} -\hspace{1mm}2$

$8x\hspace{1mm} – \hspace{1mm}6x = 4\hspace{1mm}+\hspace{1mm}2$

$2x = 6$

$x = \dfrac{6}{2} = 3$

### Example 4:

Calculate the lengths of the unknown sides of the triangle by using the perpendicular bisector theorem.

__Solution:__

From the properties of the perpendicular bisector theorem, **we know that the side** $AD = BD$.

$10x\hspace{1mm} +\hspace{1mm}5 = 15x -25$

$15x – 10x = 5\hspace{1mm}+\hspace{1mm}25$

$5x = 30$

$x = \dfrac{30}{5} = 6$

### Example 5:

Mason is standing in a playground. The playground is used for playing football, and it has a pair of goalposts. The distance between the two poles is $6$ inches. Suppose Mason was standing at point C, and he moves forward in a straight line and ends up at point M between the two poles. If the distance of one pole to point C is $-2x\hspace{1mm} +\hspace{1mm}6$ and the distance of the other pole to point C is $10x\hspace{1mm} –\hspace{1mm} 6$ inches, then calculate the distance covered by Mason from point C to M.

__Solution:__

Let us draw the figure for the given problem. When Mason moves in a straight line from point C to M, **it forms a perpendicular bisector on the two poles**. Assume one pole is X and the other is Y.

$-2x +6 = 10x – 6$

$10x + 2x = 6+6$

$12x = 12$

$x = \dfrac{12}{12} = 1$

Putting the value of “$x$” *in both equations:*

$-2 (1) \hspace{1mm}+\hspace{1mm} 6 = -2 \hspace{1mm}+ \hspace{1mm}6 = 4$ inches

$10(1) \hspace{1mm}–\hspace{1mm} 6 = 10\hspace{1mm} – \hspace{1mm}6 = 4$ inches

As M **is the mid point of XY and it divides XY equally in half**, so the length for XM and YM is equal to $3$ inches each.

Applying Pythagoras theorem to *calculate the distance covered by Mason from point C to M:*

$XC^{2} = XM^{2}\hspace{1mm} +\hspace{1mm} CM^{2}$

$CM = \sqrt{XC^{2}\hspace{1mm}- \hspace{1mm}XM^{2}}$

$CM = \sqrt{4^{2}\hspace{1mm}-\hspace{1mm} 20^{2}}$

$CM = \sqrt{16 \hspace{1mm}-\hspace{1mm} 9}$

$CM = \sqrt {7} = 2.65$ inches approx.

*Practice Questions*

- Using the properties of perpendicular bisector theorem, calculate the value of “x” for the figure given below.
- Prove that the vertex between the two equal sides in an isosceles triangle lies on the perpendicular bisector of the base.

*Answer Key*

1.

From the properties of perpendicular bisector theorem, **we know that the side** $AC = BC$.

$12x \hspace{1mm}+\hspace{1mm} 4 = 8x\hspace{1mm} +\hspace{1mm}12$

$12x\hspace{1mm} –\hspace{1mm} 8x = 12\hspace{1mm} –\hspace{1mm} 4$

$4x = 8$

$x = \dfrac{8}{4} = 2$

2.

Let us draw a perpendicular from the vertex $A$ to point $M$ at the line segment $BC$. As the triangle is an isosceles, $AB$ **and** $AC$ **are equal**. So the point $A$ is equidistant from the endpoints of $BC$. By converse perpendicular bisector theorem,

$BM = CM$

Hence,** the vertex lies on the perpendicular bisector of base** $BC$.