Geometry - Poudre School District



Geometry Notes – Lesson 5.3 Name __________________________________

Concurrent Lines: _______________________________________________________________________________________.

Point of Concurrency: ___________________________________________________________________________________.

Perpendicular Bisectors of the Sides of a Triangle.

Circumcenter of a Triangle:

The Point of Concurrency of the _________________________ __________________________ of a Triangle.

Example. Point _______ is the circumcenter.

Theorem 5-6: The ________________________ ________________________ of the sides of a triangle

are concurrent at a point __________________________ from the ______________________.

Example __________ = ___________ = ___________

By using the circumcenter, you are able to draw a circle The circle is ____________________________ about the triangle.

around the triangle going through each vertex.

Finding the circumcenter.

Examples:

Find the center of the circle that you can circumscribe about the triangle with vertices

(0, 0)

(-8, 0)

(0, 6)

Find the center of the circle that you can circumscribe about the triangle with vertices

(1, 1)

(5, 1)

(1, 7)

Angle Bisectors of a Triangle.

Incenter of a Triangle:

The Point of Concurrency of the _________________________ __________________________ of a Triangle.

Example. Point _______ is the incenter.

Theorem 5-7: The ____________________ of the angles of a triangle

are concurrent at a point _____________________ from the ___________.

Example __________ = ___________ = ___________

By using the Incenter, you are able to draw a circle The circle is ____________________________ in the triangle.

Inside the triangle by measuring from the incenter to a side.

Would you find the circumcenter or the incenter?

a) The towns of Adamsville, Brooksville, and b) City planners want to locate a fountain

Cartersville want to build a library equidistant from three straight roads

that is equidistant from the three towns. that enclose a park.

Medians of a Triangle: ___________________________________________________________________________________

__________________________________________________________________________________________.

Centroid of a Triangle:

The Point of Concurrency of the __________________________ of a Triangle.

Example. Point _______ is the centroid.

The centroid is also called the ____________________ _____ ____________________

of a triangle because it is the point where a triangular shape will balance.

Theorem 5-8: The distance from a vertex to the centroid is _______

the distance from each vertex to the midpoint of the opposite side.

[pic], [pic], [pic]

Also, the distance from the centroid to the opposite side is ________ the

distance from each vertex to the midpoint of the opposite side.

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Examples.

Altitudes of a Triangle:

________________________________ ________________________ from a vertex to the line containing the opposite side.

***Unlike angle bisectors and medians, an altitude can be a side or it may lie inside or outside the triangle.

Orthocenter of a Triangle:

The point of concurrency of the lines containing the _________________________ of a Triangle.

Theorem 5-9: ___________________________________________________________________________________________

Examples.

a) Is KX a median, an altitude, neither b) Is ST a median, an altitude, neither

or both? or both?

c) Is UW a median, an altitude, neither

or both?

Examples.

Is AB an angle bisector, altitude, median or perpendicular bisector?

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