FeatureLesson

GeometryGeometry

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(For help, go to Lesson 1-7.)

Lesson 5-3

1. an angle bisector

2. a perpendicular bisector of a side

3. Draw GH Construct CD GH at the midpoint of GH.

4. Draw AB with a point E not on AB. Construct EF AB.

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

Draw a large triangle. Construct each figure.

Check Skills You’ll Need

Check Skills You’ll Need

5-3

FeatureLesson

GeometryGeometry

LessonMain

1–2. 3.

4.

Solutions

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

Answers may vary. Samples given:

Check Skills You’ll Need

5-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

Warm Up

1. JK is perpendicular to ML at its midpoint K. List the congruent segments.

Find the midpoint of the segment with the given endpoints.

2. (–1, 6) and (3, 0)

3. (–7, 2) and (–3, –8) (–5, –3)

(1, 3)

FeatureLesson

GeometryGeometry

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FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

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

The point of concurrency of the three perpendicular bisectors of a triangle is the circumcenter of the triangle.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .

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

FeatureLesson

GeometryGeometry

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Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

5-3

Circumcenter Theorem

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Incenter Theorem

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

FeatureLesson

GeometryGeometry

LessonMain

Find the center of the circle that circumscribes XYZ.

Because X has coordinates (1, 1) and Z has coordinates (5, 1), XZ lies on

the horizontal line y = 1. The perpendicular bisector of XZ is the vertical line

that passes through ( , 1) or (3, 1), so the equation of the perpendicular

bisector of XZ is x = 3.

1 + 5 2

Because X has coordinates (1, 1) and Y has

coordinates (1, 7), XY lies on the vertical line x = 1.

The perpendicular bisector of XY is the horizontal line

that passes through (1, ) or (1, 4), so the equation

of the perpendicular bisector of XY is y = 4.

1 + 7 2

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

Additional Examples

5-3

Finding the Circumcenter

You need to determine the equation of two bisectors, then determine the point of intersection.

FeatureLesson

GeometryGeometry

LessonMain

(continued)

The lines y = 4 and x = 3 intersect at the point (3, 4).

This point is the center of the circle that circumscribes XYZ.

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

Quick Check

Additional Examples

5-3

FeatureLesson

GeometryGeometry

LessonMain

City planners want to locate a fountain equidistant from three

straight roads that enclose a park. Explain how they can find the

location.

Theorem 5-7 states that the bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

The city planners should find the point of concurrency of the angle bisectors of the triangle formed by the three roads and locate the fountain there.

The roads form a triangle around the park.

Lesson 5-3

Concurrent Lines, Medians, and AltitudesConcurrent Lines, Medians, and Altitudes

Additional Examples

5-3

Real-World Connection

Quick Check

FeatureLesson

GeometryGeometry

LessonMain

The point of concurrency of the perpendicular bisectors of the sides of a triangle.

CircumcenterCircumcenter

FeatureLesson

GeometryGeometry

LessonMain

CircumcenterCircumcenter

The circumcenter is equidistant from each vertex of the triangle.

FeatureLesson

GeometryGeometry

LessonMain

The point of concurrency of the three angles bisectors of the triangle.

IncenterIncenter

FeatureLesson

GeometryGeometry

LessonMain

The incenter is equidistant from the sides of a triangle.

IncenterIncenter

FeatureLesson

GeometryGeometry

LessonMain

The incenter is equidistant from the sides of a triangle.

IncenterIncenter