Concurrent Lines, Medians, and Altitudes

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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Check Skills Youll Need
(For help, go to Lesson 1-7.)
Draw a large triangle. Construct each figure.
Check Skills Youll Need
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Check Skills Youll Need
Solutions
Answers may vary. Samples given
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Concurrent Lines, Medians, and Altitudes
Lesson 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)
(1, 3)
(5, 3)
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Concurrent Lines, Medians, and Altitudes
Lesson 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.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
The point of concurrency of the three
perpendicular bisectors of a triangle is the
circumcenter of the triangle.
The circumcenter can be inside the triangle,
outside the triangle, or on the triangle.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
The circumcenter of ?ABC is the center of its
circumscribed circle. A circle that contains all
the vertices of a polygon is circumscribed about
the polygon.
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Concurrent Lines, Medians, and Altitudes
Lesson 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.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
The incenter is the center of the triangles
inscribed circle. A circle inscribed in a polygon
intersects each line that contains a side of the
polygon at exactly one point.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
The perpendicular bisectors of the sides of a
triangle are concurrent at a point equidistant
from the vertices.
The bisectors of the angles of a triangle are
concurrent at a point equidistant from the sides.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Additional Examples
Finding the Circumcenter
You need to determine the equation of two ?
bisectors, then determine the point of
intersection.
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Additional Examples
Quick Check
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Additional Examples
Real-World Connection
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.
Quick Check
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Circumcenter

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

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Circumcenter

• The circumcenter is equidistant from each vertex
  of the triangle.

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Incenter

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

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Incenter

• The incenter is equidistant from the sides of a
  triangle.

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Incenter

• The incenter is equidistant from the sides of a
  triangle.