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Example 1 A: Applying the Perpendicular Bisector Theorem and Its Converse Example 1 A: Applying the Perpendicular Bisector Theorem and Its Converse

Example 1 B: Applying the Perpendicular Bisector Theorem and Its Converse Example 1 B: Applying the Perpendicular Bisector Theorem and Its Converse

Example 1 C: Applying the Perpendicular Bisector Theorem and Its Converse Example 1 C: Applying the Perpendicular Bisector Theorem and Its Converse

Check It Out! Example 1 a Check It Out! Example 1 a

Check It Out! Example 1 b Check It Out! Example 1 b

Remember that the distance between a point and a line is the length of Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line.

Based on these theorems, an angle bisector can be defined as the locus of Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.

Example 2 A: Applying the Angle Bisector Theorem Example 2 A: Applying the Angle Bisector Theorem

Example 2 B: Applying the Angle Bisector Theorem and , bisects of the Angle Example 2 B: Applying the Angle Bisector Theorem and , bisects of the Angle Bisector Theorem.

Example 2 C: Applying the Angle Bisector Theorem by the Converse of the Angle Example 2 C: Applying the Angle Bisector Theorem by the Converse of the Angle Bisector Theorem.

Check It Out! Example 2 a Check It Out! Example 2 a

Check It Out! Example 2 b Check It Out! Example 2 b

Example 3: Application Example 3: Application

Check It Out! Example 3 Check It Out! Example 3

Example 4: Writing Equations of Bisectors in the Coordinate Plane The perpendicular bisector of Example 4: Writing Equations of Bisectors in the Coordinate Plane The perpendicular bisector of is perpendicular to at its midpoint.

Example 4 Continued mdpt. of = Example 4 Continued mdpt. of =

Example 4 Continued Since the slopes of perpendicular lines are opposite reciprocals, the slope Example 4 Continued Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is

Example 4 Continued Example 4 Continued

Example 4 Continued Example 4 Continued

Check It Out! Example 4 The perpendicular bisector of is perpendicular to at its Check It Out! Example 4 The perpendicular bisector of is perpendicular to at its midpoint.

Check It Out! Example 4 Continued Check It Out! Example 4 Continued

Check It Out! Example 4 Continued Since the slopes of perpendicular lines are opposite Check It Out! Example 4 Continued Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is .

Check It Out! Example 4 Continued Check It Out! Example 4 Continued

Since a triangle has three sides, it has three perpendicular bisectors. When you construct Since a triangle has three sides, it has three perpendicular bisectors. When you construct the perpendicular bisectors, you find that they have an interesting property.

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

The circumcenter can be inside the triangle, outside the triangle, or on the triangle. The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

Example 1: Using Properties of Perpendicular Bisectors Example 1: Using Properties of Perpendicular Bisectors

Check It Out! Example 1 a Check It Out! Example 1 a

Check It Out! Example 1 b Check It Out! Example 1 b

Check It Out! Example 1 c Check It Out! Example 1 c

Example 2: Finding the Circumcenter of a Triangle Example 2: Finding the Circumcenter of a Triangle

Example 2 Continued Example 2 Continued

Example 2 Continued Example 2 Continued

Check It Out! Example 2 Check It Out! Example 2

Check It Out! Example 2 Continued Check It Out! Example 2 Continued

Check It Out! Example 2 Continued Check It Out! Example 2 Continued

The distance between a point and a line is the length of the perpendicular The distance between a point and a line is the length of the perpendicular segment from the point to the line.

Unlike the circumcenter, the incenter is always inside the triangle. Unlike the circumcenter, the incenter is always inside the triangle.

Example 3 A: Using Properties of Angle Bisectors Example 3 A: Using Properties of Angle Bisectors

Example 3 B: Using Properties of Angle Bisectors Example 3 B: Using Properties of Angle Bisectors

Check It Out! Example 3 a Check It Out! Example 3 a

Check It Out! Example 3 b Check It Out! Example 3 b

Example 4: Community Application Let the three towns be vertices of a triangle. By Example 4: Community Application Let the three towns be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. Draw the triangle formed by the three buildings. To find the circumcenter, find the perpendicular bisectors of each side. The position for the library is the circumcenter.

Check It Out! Example 4 Check It Out! Example 4

Lesson Quiz: Part I 17 3 Lesson Quiz: Part I 17 3

Lesson Quiz: Part II Lesson Quiz: Part II