1. Bisectors, Medians, and Altitudes Section 5-1
Agenda:
11/30/11
Do Now
Problem involving Isosceles and Equilateral Triangles
Review Do Now
Vocabulary:
Perpendicular bisector,
angle bisector,
distance from a point to a line
Mini Lesson:
Using properties of Perpendicular bisectors and Angle Bisectors to solve problems
Independent /Group work
Practice problems
Share-out
Discussion of answers
Wrap-Up/ Summary
Writing Exercise
Lesson Quiz
Homework
Review Class Notes,
Castle Learning
Relationships in Triangles
Bisectors, Medians, and Altitudes
Section 5-1
Mrs. Padilla
Geometry Fall 2011

2. Objectives (W.O.W.) To identify and use perpendicular
bisectors & angle bisectors in triangles
To identify and use medians & altitudes in triangles

3. Vocabulary Perpendicular Bisectors Angle Bisectors Locus Equidistant
Medians
Altitudes
Points of Concurrency

4. AIM:. How do we use properties of Perpendicular
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Do Now(5 mins)

5. Perpendicular bisector in Triangles
Perpendicular bisector: A line or line
segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.
Theorem 5-1-: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
Theorem 5-2-: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

6. Perpendicular bisector theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
If C is on the
perpendicular
bisector of AB,
then CA = CB. IF C M B A
THEN C ~ M B A

7. Converse of the Perpendicular bisector theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.

8. Perpendicular bisector, cont…
For every triangle there are 3 perpendicular bisectors
The 3 perpendicular bisectors intersect in a common
point named the circumcenter.
In the picture to the right
point K is the circumcenter.

9. Angle bisectors of a Triangle
Angle bisector of a triangle:
A segment that bisects an angle of a triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle.
Theorem 5-3: Any point on the bisector of an angle is equidistant from the sides of the angle.
Theorem 5-4: Any point on or in the interior of an angle and equidistant from the sides of an angle, lies on the bisector of the angle.

10. Angle bisector theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
IF THEN
If m< 1 = m< 2, then BC = BD. C B B 1 1 A 2 A 2 D ~

11. Angle bisector, cont… For every triangle there are 3 angle bisectors.
The 3 angle bisectors intersect in a common point named the incenter
In the picture to the right, point I is the incenter.

12. Median and Altitude
Median: A segment that connects a vertex of a triangle to the midpoint of the side opposite to that vertex. Every triangle has three medians.
Altitude: A segment that has an endpoint at
a vertex of a triangle and the other on the line
opposite to that vertex, so that the segment is
perpendicular to this line. Do example 1, page 239
Altitudes of
an obtuse
triangle
Altitudes of
a right triangle

13. Median A line segment whose endpoints are a vertex of a
triangle and the midpoint of the side opposite the
vertex.
In the picture to the right, the blue line segment is the median.

14. Median, cont… For every triangle there are 3 medians
The 3 medians intersect in a common point named the centroid
In the picture to the right, point O is the centroid.

15. Altitudes
A line segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
In the picture above, ∆ABC is an obtuse triangle & ∠ACB is the obtuse angle. BH is an altitude.

16. Altitudes, cont… For every triangle there are 3 altitudes
The 3 altitudes intersect in a common point called
the orthocenter.
In the picture to the right, point H is the orthocenter.

17. Points of Concurrency
Concurrent Lines
3 or more lines that intersect at a common point
Point of Concurrency
The point of intersection when 3 or more lines intersect.
Type of Line Segments Point of Concurrency
Perpendicular Bisectors Circumcenter
Angle Bisectors Incenter
Median Centroid
Altitude Orthocenter

18. Points of Concurrency, cont…
Facts to remember:
The circumcenter of a triangle is equidistant from the
vertices of the triangle.
Any point on the angle bisector is equidistant from the sides of the angle (Converse of #3)
Any point equidistant from the sides of an angle lies on
the angle bisector. (Converse of #2)
The incenter of a triangle is equidistant from each side of the triangle.
The distance from a vertex of a triangle to the centroid is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.

19. Points of Concurrency, cont…

20. Example: Using Perpendicular bisector
Use the diagram to find AB.
In the diagram, AC is the perpendicular
bisector of DB. Therefore AB = AD
8x = 5x + 12
3x = 12
x = 4
Since you were asked for AB, not just x:
AB = 8x = 8 • 4 = 32 C D B
5x + 12 8x A

21. AIM:. How do we use properties of Perpendicular
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Example

22. AIM:. How do we use properties of Perpendicular
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Examples
Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?

23. AIM:. How do we use properties of Perpendicular
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Examples
Does the information given in the diagram allow you to conclude that P is on the angle bisector of angle A? A A A

24. How can you tell if a ray or line segment is an angle bisector?
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Summary
How can you tell if a ray or line segment is an angle bisector?
How can you tell if a ray or line segment is a perpendicular bisector?

25. Facts to Remember and Memorize!
1. Perpendicular Bisectors
2. Angle Bisectors
3. Medians
4. Altitudes
1. …form right angles AND  lines segments
2. …form 2  angles
3. …form 2  line segments
4. … form right angles

26. Lesson Quiz 1. Does D lie on the perpendicular bisector of
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Lesson Quiz
1. Does D lie on the perpendicular bisector of
Draw the diagram and answer the question

27. Review Class Notes Sec 5.1N and Sec 4.8R HomeWork
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
HomeWork
Review Class Notes
Sec 5.1N and Sec 4.8R

28. The End
(Finally!)
Oh yeah! Do homework tonight and STUDY these notes that you just took on Section 5-1!