1. 5-1 Bisectors, Medians, and Altitudes

2. Perpendicular Bisector- line/segment that has the midpoint of the side and is perpendicular to it
Altitude D
D is the midpoint of BC.

3. Theorems
5.1: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
5.2: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

4. Point of concurrency: The point of intersection.
Concurrent lines: When three or more lines intersect at a common point.
Point of concurrency: The point of intersection.
Circumcenter: The point of concurrency of
the perpendicular bisectors of a triangle
Theorem 5.3: The circumcenter of a triangle is equidistant from the
vertices of the triangle.
Point of Concurrency
Called circumcenter if the lines are perpendicular bisectors.

5. Points on Angle Bisectors
Theorem 5.4: Any point on the angle bisector is equidistant from the sides of the angle.
Theorem 5.5: Any point equidistant from the sides of an angle lies on the angle bisector.

6. Incenter
Incenter: The intersection point of the angle bisectors of a triangle.
Theorem 5.6: (Incenter Theorem): The incenter of a triangle is equidistant from each side of the triangle.

7. Median: A segment whose endpoints are a vertex of the triangle and the midpoint of the side opposite of the vertex.
E N G F
In  EFG, FN is a median. Find EN if EG = 11
FN is the median. So, N is the midpoint of EG. Since EG = 11, EN = ½ · 11 or 5.5.

8. Centroid: When the medians of a triangle intersect at a common point.K Q M R P X J
X is the centroid of  JKM
JR, KP, and MQ are concurrent
Centroid Theorem (5.7): The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

9. Check your progress
Try page 272 #2 (check your progress)

10. Altitudes
Altitude: A perpendicular segment with one endpoint at a vertex and the other on the opposite side.
Orthocenter: The point of concurrency of the altitudes. A B C
Altitude
Orthocenter

11. Altitude Examples Altitude Right angle
The altitude is a side of the triangle.
Acute angle
The altitude is inside the triangle.
Altitude
Altitude
Obtuse angle
The altitude is outside the triangle.

12. EXAMPLE
 In RST, RP and SQ are medians. If RQ=7x-1, SP=5x-4, and QT=6x+9, find PT.
 Since RP and SQ are medians, Q and P are midpoints. Use the given values for RQ and QT to first solve for x.
RQ = QT
7x - 1 = 6x + 9
7x = 6x
7x = 6x + 10
7x - 6x = 6x x
x = 10
 Next, use the values of x and SP to find PT.
SP = PT
5x - 4 = PT
5(10) – 4 = PT
46 = PT R S T P Q

13. In RST, RQ, SP, and OT are medians. Find AQ if RA = 6. Since RA = 6,
RA = 2/3 (RQ)
6 = 2/3 (RQ)
(3/2) (6) = RQ
9 = RQ so AQ = 3
If OA = 2.4, what is OT?
Since OA = 2.4, AT = 2.4 x 2 = 4.8
So OT = = 7.2 R S T O P Q A
Use the Centroid Theorem (5.7): The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median.

14. Y Z X E The line containing YE is the perpendicular bisector of XZ.
YE is an altitude.
E is the midpoint of XZ.