Transcript Concurrent Lines, Medians, and Altitudes

Concurrent Lines, Medians, and Altitudes
Lesson 5-3
Geometry
Check Skills You’ll Need
(For help, go to Lesson 1-7.)
Draw a large triangle. Construct each figure.
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.
Check Skills You’ll Need
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Concurrent Lines, Medians, and Altitudes
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Check Skills You’ll Need
Solutions
Answers may vary. Samples given:
1–2.
3.
4.
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Concurrent Lines, Medians, and Altitudes
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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)
(1, 3)
3. (–7, 2) and (–3, –8)
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(–5, –3)
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Concurrent Lines, Medians, and Altitudes
Lesson 5-3
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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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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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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.
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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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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.
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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.
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Additional Examples
Finding the Circumcenter
Find the center of the circle that circumscribes
XYZ.
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, 1 + 7 ) or (1, 4), so the equation
2
of the perpendicular bisector of XY is y = 4.
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 + 5 , 1) or (3, 1), so the equation of the perpendicular
2
bisector of XZ is x = 3. You need to determine the equation of two 
bisectors, then determine the point of intersection.
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(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.
Quick Check
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Additional Examples
Real-World Connection
City planners want to locate a fountain equidistant from three
straight roads that enclose a park. Explain how they can find the
location.
The roads form a triangle around the park.
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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Geometry
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.
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