Concurrent Lines Geometry Mrs. King Unit 4, Day 7.
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Concurrent Lines Geometry Mrs. King Unit 4, Day 7
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Transcript of Concurrent Lines Geometry Mrs. King Unit 4, Day 7.
Concurrent Lines
Geometry
Mrs. King
Unit 4, Day 7
Definition
• Concurrent – when three or more lines intersect in one point
Definition
• Point of Concurrency – the point at which lines are concurrent
On your triangle, complete a perpendicular bisector on all three sides.
On your triangle, complete a perpendicular bisector on all three sides.
On your triangle, complete a perpendicular bisector on all three sides.
Draw a Triangle
• What do you notice about the perpendicular bisectors of a triangle?
• What about the relationship between the point of concurrency and the vertices?
More Theorems
• Theorem 4-16: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Theorem 4-16 Continued:
Q
• The point of concurrency (point C) is called the circumcenter.
• The circumcenter is equidistant from all the vertices
• The circle is circumscribed around the triangle
S
R
On your triangle, complete a angle bisector on all three angles.
Draw a Triangle
• What do you notice about the angle bisectors of a triangle?
• What about the relationship between the point of concurrency and the sides?
More Theorems
• Theorem 4-17: The bisector of the angles of a triangle are concurrent at a point equidistant from the sides
Theorem 4-17 Continued:
• The point of concurrency (point I) is called the incenter.
• The incenter is equidistant from all the sides
• The circle is inscribed in the triangle
U
T
V
I
Definition
• Altitude of a triangle – a perpendicular segment from a vertex to the line containing the side opposite the vertex.
Altitudes
• An altitude of a triangle may lie outside the triangle:
More Theorems
• Theorem 4-18: The lines that contain the altitudes of a triangle are concurrent.
Definition
• Median of a triangle – a segment whose endpoints are a vertex and the midpoint of the side opposite the vertex.
Are Medians Concurrent?
• Draw a triangle and then draw all three medians.
• Are they concurrent?
More Theorems
• Theorem 4-19: The medians of a triangle are concurrent.
Summary
Perpendicular Bisectors Angle Bisectors
MediansAltitudes
Homework
Work Packet: Concurrent Lines
