Chapter 5 Vocabulary and Notes
Big Ideas: Relationships with Triangles
1. Use properties of special segments in triangles
2. Use triangle inequalities to determine what triangles are possible
3. Extend methods for justifying and proving relationships
Key Vocabulary:
5.1 Midsegment Theorem and Coordinate Proof
Objective:
 Use the properties of midsegments and Theorem 5.1—Midsegment Theorem
to find lengths
 Write coordinate proofs by placing a figure in the coordinate plane, assigning
coordinates to the vertices, and then using the midpoint, distance, and/or
slope formulas
Essential Question: How do you write a coordinate proof?
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5.2 Use Perpendicular Bisectors
Objective:
 Define perpendicular bisector and equidistant, and explain how they apply
to Theorem 5.2—Perpendicular Bisector Theorem and Theorem 5.3—
Converse of the Perpendicular Bisector Theorem
 Use the Perpendicular Bisector Theorem to find segment measures
 Define concurrency and identify the point of concurrency—explain how the
point of concurrency has a special property
 Use Theorem 5.4—Concurrency of Perpendicular Bisectors of a Triangle to
solve for segment measures
 Define circumcenter and explain how this point P depends on the type of
triangle
Essential Question: How do you find the point of concurrency of the
perpendicular bisectors of the sides of a triangle?
Activity: Fold the Perpendicular Bisectors of a Triangle—text pg. 304
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The perpendicular bisector of a side of a triangle can be referred to as a
perpendicular bisector of the triangle
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5.3 Use Angle Bisectors of Triangles
Objective:
 Use angle bisectors to find distance relationships
 Apply Theorem 5.5 (Angle Bisector Theorem) and Theorem 5.6
(Converse of the Angle Bisector Theorem) to solve for the distance
(length) of segments
 Apply Theorem 5.7 (Concurrency of Angle Bisectors of a Triangle) to
find angle measures
Essential Question: What can you conclude that a point is on the bisector
of an angle?
What is an angle bisector?
*Remember—in geometry, distance means the shortest length between
two objects
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Look at these examples. In example 1, you are using the Angle Bisector
Theorem to solve for an unknown angle measure and in example 2, you
are using the theorem to solve for a missing distance
Example 1
Example 2
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5.4 Use Medians and Altitudes
 Use Theorem 5.8 (Concurrency of Medians of a Triangle) to find the
coordinates of the centroid of a triangle
 Use Theorem 5.9 (Concurrency of Altitudes of a Triangle) to identify
the orthocenter of a triangle
Essential Question: How do you find the centroid of a triangle?
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Think of it as having “three” different bases—I turn the triangle to identify the
altitudes! Sometimes you must “extend” the side opposite each vertex in order
to identify the altitude and therefore, the point at which the three altitudes
intersect.
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