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FeatureLesson
GeometryGeometry
LessonMain
(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
LessonMain
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
LessonMain
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