1. Bisectors, Medians, and Altitudes
Skill 28

2. Objective
HSG-C.3/10: Students are responsible for using properties of bisectors, medians and altitudes. Also, using these properties to understand problems.

3. Definitions
When three or more lines intersect at one point they are called, concurrent.
The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of a triangle.
Using the circumcenter as the center of a circle the circle will contain each vertex and the circle is circumscribed about the triangle.

4. Definitions
The point of concurrency of the angle bisectors of a triangle is called the incenter of a triangle.
A circle with center at the incenter of a triangle is inscribed in the triangle.
A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side.

5. Definitions
The point of concurrency of the medians of a triangle is the centroid of a triangle.
An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side.
The point of concurrency of the altitudes of a triangle is the orthocenter of a triangle.

6. Theorem 28: Concurrency of Perp. Bisectors Thm.
The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.
Circumcenter

7. Theorem 29: Concurrency of Angle Bisectors Thm.
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle.
incenter

8. Theorem 30: Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is two-thirds the distance from each vertex to the midpoint of the opposite side.
Centroid

9. Theorem 31: Concurrency of Altitudes Theorem
The lines that contain the altitudes of a triangle are concurrent.
Orthocenter

10. Example 1; Using the Circumcenter
A town planer wants to locate a new fire station equidistant from the elementary, middle, and high schools. Where should the station be located? Explain.
Need to find the circumcenter, b/c it is equidistant from each vertex.
Find the perpendicular bisector of 𝑯𝑴 E H M
Find the perpendicular bisector of 𝑬𝑴
Find the perpendicular bisector of 𝑯𝑬 F
The fire station should be placed at point F.

11. Example 2; Identifying and Using the Incenter
𝐺𝐸=2𝑥−7 and 𝐺𝐹=𝑥+4. What is 𝐺𝐷?
G is the incenter b/c all of the angle bisectors meet at G.
GD, GE, and GF are the distance from G to each side, by definition of distance
𝑮𝑫=𝑮𝑬=𝑮𝑭, by concurrency of Angle Bisector Theorem A B C D E F G
𝟐𝒙+𝟕=𝒙+𝟒
𝒙=𝟏𝟏
𝑮𝑫=𝑮𝑭=𝒙+𝟒
𝑮𝑫= 𝟏𝟏 +𝟒
𝑮𝑫=𝟏𝟓

12. Example 3; Identifying the Length of the Median
Suppose 𝑋𝐴=8, find the length of 𝑋𝐵 .
The distance from a vertex to the centroid of a triangle is 2/3 the total length. X Y Z A B C
𝑿𝑨= 𝟐 𝟑 𝑿𝑩
𝟖= 𝟐 𝟑 𝑿𝑩
𝟏𝟐=𝑿𝑩

13. Example 4; Identifying Medians and Altitudes
a) For ∆𝑃𝑄𝑆, is 𝑃𝑅 a median, an altitude, or neither? Explain. R S Q P T
𝑷𝑹 is going from a vertex to the opposite side, and is perpendicular to that side.
Altitude
b) For ∆𝑃𝑄𝑆, is 𝑄𝑇 a median, an altitude, or neither? Explain.
T is the midpoint of 𝑻𝑺 which is opposite of vertex Q.
Median

14. #28: Bisectors, Medians, and Altitudes
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