Perpendicular Bisector and Circumcenter

Perpendicular Bisectors

Geometry Honors

Ms. Hogan

Math Spoken Here!

Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

Equidistant: A point is equidistant from two figures if it is the same distance from each figure.

Note that the red line is equidistant from the endpoints of c and is perpendicular to side c.

Need to know...

In a plane:

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

If a point is equidistant from the endpoints of a segment then it is on the perpendicular bisector of the segment.

Take Notice:

Did you notice that when the perpendicular bisector passes through the vertex point and the base, then the legs are congruent?

In an isosceles triangle a perpendicular bisector that passes through the base will always pass through the vertex point leaving two congruent legs.

Look closely at the image.

Multiple Choice

Find HG

• 28
• 14
• 41
• 7

Multiple Choice

• JM = 5

• JM = 10

• JM = 12

• JM = 24

Multiple Select

Select every true statement below.

• BC = 10 ft.

• AD = 36 ft.

• BE = 10 ft.

• DE = 36 ft.

Take note:

The perpendicular bisector of a side of a triangle does not always pass through the opposite vertex.

Math Spoken Here!

Concurrent: When three or more lines rays or segments intersect in the same point.

Point of Concurrency: The point of intersection of the lines, rays, or segments.

Math Proven Here!

Given: Lines ℓ, m, and n are the perpendicular bisectors of AB , BC , and _ AC , respectively. P is the intersection of ℓ, m, and n. Prove: PA = PB = PC

​

​P is the intersection of ℓ, m, and n. Since P lies on the perpendicular bisector of AB , PA = PB by the Perpendicular Bisector Theorem. Similarly, P lies on the perpendicular bisector of BC , so PB = PC. Therefore, PA = PB = PC by the Transitive Property of Equality.

An example of the use of concurrency of perpendicular bisectors.

Math Spoken Here!

Circumcenter: The point of concurrency of the three perpendicular bisectors of a triangle.

Circumcenter:

The circumcenter (call it P) is equidistant from the three vertices, so P is the center of a circle that passes through all three vertices.

As shown, the location of P depends on the type of triangle. The circle with center P is said to be circumscribed about the triangle.

Multiple Choice

Point P is the circumcenter, which segment is equal to PB

• DA
• AC
• PC
• PF

Multiple Choice

Name the point of concurrency shown.

• Circumcenter
• Incenter
• Supercenter
• Neither

Great job!

More information and practice can be found in your HMH Textbook L8.1