The concept of a perpendicular bisector is a line segment that divides another line segment into two equal parts at a right angle (90°). This fundamental principle has practical applications in various fields such as architecture and surveying, making it a valuable concept to understand in geometry.

To grasp the concept of a perpendicular bisector, it is helpful to visualize it on a Cartesian plane. The perpendicular bisector intersects the midpoint of two points, A (x1, y1) and B (x2, y2), that are on the line segment. This point of intersection is represented by the coordinates M (xm, ym). Notably, the distance from the midpoint to either point A or B is equal, signifying that AM = BM.

The equation for a line passing through points A and B can be written as y = m1x + c, with m1 being the slope of the line. Similarly, the equation for the perpendicular bisector of this line can be expressed as y = m2x + d, where m2 is the slope of the perpendicular bisector.

One key aspect to note is that the product of the slopes of two perpendicular lines is always -1. In other words, m1 x m2 = -1. This fact is crucial in determining the equation of a perpendicular bisector.

Given two points A (x1, y1) and B (x2, y2), the following method can be used to determine the equation of the perpendicular bisector that intersects the midpoint between A and B.

- Step 1: Use the Midpoint Formula to find the coordinates of the midpoint (xm, ym): (x1 + x2)/2, (y1 + y2)/2
- Step 2: Calculate the slope of the line segment connecting A and B using the Gradient Formula: (y2 - y1)/(x2 - x1)
- Step 3: Determine the slope of the perpendicular bisector using the formula m2 = -1/m1.
- Step 4: Plug in the values for the midpoint and slope into the Equation of a Line Formula, y - ym = m2(x - xm), to obtain the equation of the perpendicular bisector.

For example, let's find the equation of the perpendicular bisector for the line segment connecting the points (9, -3) and (-7, 1).

**Solution:**

- Assign (x1, y1) = (9, -3) and (x2, y2) = (-7, 1).
- Use the Midpoint Formula to find the coordinates of the midpoint (xm, ym): (1, -1).
- The slope of the line segment is m1 = (1 - (-3))/(-7 - 9) = -1/4.
- The slope of the perpendicular bisector is m2 = -1/m1 = -4.
- Substitute the values into the Equation of a Line Formula, y = 4x - 5, to determine the equation for the perpendicular bisector.

The Perpendicular Bisector Theorem states that any point on the perpendicular bisector is equidistant from both endpoints of a line segment. In simpler terms, a point is equidistant from a set of coordinates if the distance between the point and each coordinate in the set is the same.

Using the SAS Congruence rule, the Perpendicular Bisector Theorem can be proven:

*SAS Congruence:* If two sides and an included angle of one triangle are equal to two sides and an included angle of another triangle, the two triangles are congruent.

Referencing the diagram above, we notice that triangles XAM and YAM have:

- XM = YM as M is the midpoint
- AM = AM (shared side)
- ∠XMA = ∠YMA = 90°

Thus, according to the SAS Congruence rule, triangles XAM and YAM are congruent. By using the Converse of the Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we can conclude that A is equidistant from points X and Y. This means that XA = YA, providing evidence for the Perpendicular Bisector Theorem.

The Converse of the Perpendicular Bisector Theorem states that if a point is equidistant from the endpoints of a line segment in the same plane, then that point lies on the perpendicular bisector of the line segment. This concept is valuable in geometry and can assist in determining the perpendicular bisector's equation when given a point and two other points on the line segment.

Let's say we have the triangle XYZ shown below. If the perpendicular bisector of the line segment BZ is XA for the triangle XBZ, find the length of side XZ if XB = 17 cm and AZ = 6 cm.

To better understand the concept, refer to the diagram provided below.

The Converse of the Perpendicular Bisector Theorem states that if the point XP is equal to YP, then the point P lies on the perpendicular bisector of the line segment XY. This can be proven by considering the given diagram and information.

**Proof:**

Given: XA = YA

To Prove: XM = YM

Construct a perpendicular line from point A to the line segment XY, intersecting at point M. This creates two triangles, XAM and YAM. By comparing these triangles, we can see that XA = YA (given), AM = AM (shared side), and ∠XMA = ∠YMA = 90o. Using the Side-Angle-Side Congruence rule, it can be concluded that triangles XAM and YAM are congruent. As point A is equidistant from both X and Y, it lies on the perpendicular bisector of the line XY. Therefore, XM = YM and M is also equidistant from X and Y.

**Example:**

Given the triangle XYZ below, determine the length of the sides AY and AZ if XZ = XY = 5 cm. The line AX intersects the line segment YZ at a right angle at point A.

As XZ = XY = 5 cm, it is evident that A must be located on the perpendicular bisector of YZ, according to the Converse of the Perpendicular Bisector Theorem. Thus, AY = AZ. Solving for x, we get 2x-1 = x+1 ⇒ 2x-x = 1+1 ⇒ x = 2. Substituting this value into the equation, we obtain AY = 2x-1 ⇒ 2(2)-1 ⇒ AY = 3. Therefore, AY = AZ = 3 cm.

The perpendicular bisector of a triangle is a line segment that runs from one side of the triangle to the opposite vertex. This line is perpendicular to the side and passes through the midpoint of the triangle. Every triangle has three perpendicular bisectors, as it has three sides. The point where all three perpendicular bisectors intersect is known as the circumcenter of the triangle.

The circumcenter is the point of concurrency of the three perpendicular bisectors of any given triangle. A point where three or more distinct lines intersect is called a point of concurrency. Similarly, three or more lines are said to be concurrent if they pass through an identical point. The illustration below depicts this concept, where P is the circumcenter of the given triangle.

**Circumcenter Theorem:**

The vertices of a triangle are equidistant from the circumcenter. In other words, in a triangle ABC, if the perpendicular bisectors of AB, BC, and AC intersect at point P, then AP = BP = CP.

**Proof:**

Consider the triangle ABC in the diagram above. The perpendicular bisectors of line segments AB, BC, and AC are given. The perpendicular bisector of AC and BC intersect at point P. We want to show that point P lies on the perpendicular bisector of AB and is equidistant from A, B, and C. Let us examine the line segments AP, BP, and CP.

According to the Perpendicular Bisector Theorem, any point on the perpendicular bisector is equidistant from both endpoints of a line segment. Therefore, AP = CP and CP = BP. Using the transitive property, AP = BP. The transitive property states that if A = B and B = C, then A = C. As per the Converse of the Perpendicular Bisector Theorem, any point equidistant from the endpoints of a segment lies on the perpendicular bisector. Therefore, P lies on the perpendicular bisector of AB. As AP = BP = CP, it can be concluded that P is equidistant from A, B, and C.

Suppose a triangle ABC is formed on a Cartesian graph by points A, B, and C. To find the circumcenter of triangle ABC, follow the steps outlined below.

- Evaluate the midpoint of two sides of the triangle.
- Find the slope of the two chosen sides.
- Calculate the slope of the perpendicular bisector of the two chosen sides.
- Determine the equation of the perpendicular bisector of the two chosen sides.
- Equate the two equations from step 4 to find the x-coordinate.
- Substitute the found x-coordinate into one of the equations from step 4 to determine the y-coordinate.
- Finding the Circumcenter of a Triangle
- Let's determine the coordinates of the circumcenter for triangle XYZ using its given vertices: X(-1, 3), Y(0, 2), and Z(-2, -2).
- To begin, we will first plot the triangle on a graph.
- Next, we need to find the midpoints of XY and XZ, which are (-0.5, 2.5) and (-1.5, 0.5) respectively. The slope of XY is -1 and the slope of XZ is 2. By finding the perpendicular bisector of XY, with a slope of 1 and an equation of y = x + 3, and the perpendicular bisector of XZ, with a slope of -0.5 and an equation of y = -0.5x, we can equate the two equations and solve for x. This results in x = -1. Substituting this value into one of the equations, we find that y = 2. Therefore, the circumcenter has coordinates (-1, 2).
- Finding the Perpendicular Bisector of XY
- The midpoint of a line segment is found using the formula (xXY, yXY) = (x1 + x2)/2, (y1 + y2)/2. Applying this formula to XY, we get (-1+0)/2, (3+2)/2 ⇒ (xXY, yXY) = (-1/2, 5/2).
- The slope of XY is found by using the formula m = (y2-y1)/(x2-x1). In this case, the slope of XY is (2-3)/(0-(-1)) ⇒ mXY = -1/1 ⇒ mXY = -1.
- The slope of the perpendicular bisector of XY is the negative reciprocal of the slope of XY, which is 1. Therefore, the equation of the perpendicular bisector is y = x + 3.
- Finding the Perpendicular Bisector of XZ
- Using the same formulas as before, the midpoint of XZ is (-1+(-2))/2, (3+(-2))/2 ⇒ (xXZ, yXZ) = (-3/2, 1/2). The slope of XZ is calculated as (-2-3)/((-2)-(-1)) ⇒ mXZ = -5/(-1) ⇒ mXZ = 5. Therefore, the slope of the perpendicular bisector of XZ is -0.5, and its equation is y = (-1/5)x + 2.
- Equating the Perpendicular Bisectors
- In order to find the coordinates of the circumcenter, we need to set the equations of the perpendicular bisectors of XY and XZ equal to each other. This can be achieved by equating the x and y values of the two equations. The x-coordinate is calculated as x + 3 = (-1/5)x + 2 ⇒ x + (1/5)x = 2 - 3 ⇒ (6/5)x = -1 ⇒ x = -1 * (5/6) ⇒ x = -5/6. Similarly, the y-coordinate can be found by solving for y in one of the equations. In this case, y = x + 3 ⇒ y = (-5/6) + 3 ⇒ y = 2 + (3/6) ⇒ y = 2 + 1/2 ⇒ y = 5/2. Thus, the circumcenter is located at (-5/6, 5/2).
- The Angle Bisector Theorem
- The angle bisector theorem states that if a point lies on the bisector of an angle, it is equidistant from the sides of the angle.

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