1. Bellringer Use the figure at the right for Exercises 1–4.
1. What is the relationship between LN and MO?
Perpendicular bisector
2. What is the value of x?
X=10
3. Find LM. 4. Find LO.
3. LM= 50
4.LO=50

2. Bellwork- MA.912.G.1.3
In the figure below, AB is parallel to DC. Which of the following statements about the figure must be true?
A. m∠DAB + m∠ABC = 180°
B. m∠DAB + m ∠ CDA = 180°
C. ∠BAD is congruent to ∠ADC
D. ∠ADC is congruent to ∠ABC
Highlands Park is located between two parallel streets, Walker Street and James Avenue. The park
faces Walker Street and is bordered by two brick walls that intersect James Avenue at point C, as shown below
What is the measure, in degrees, of ∠ACB, the angle formed by the park’s two brick walls?

3. 5-3 Bisectors in Triangles and 5.4 Medians and Altitudes
Geometry Chapter 5 Relationships within Triangles

4. Review
A point on the perpendicular bisector is equidistant to the endpoints of the segment.
A point on the angle bisector is also equidistant to the angle sides.

5. Lesson Purpose Objective Essential Question
To identify properties of perpendicular and angle bisectors.
How is the intersection of angle bisectors in a triangle different from the intersection of perpendicular bisectors?
The angle bisectors in a triangle meet at a single point that is equidistant from the sides of the triangle. This point is the incenter of an inscribed circle instead of circumscribed about the triangle .

6. Concurrency of Perpendicular Bisector Theorem
When three or more lines intersect at one point
The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.
Point of Concurrency of a Perpendicular bisectors of a triangle

7. Practice Example RE=AE 3(x+3)=21 3x+9=21 3x=21-9→3x=12 X=4 AN=KN
Solution
RE=AE
3(x+3)=21
3x+9=21
3x=21-9→3x=12
X=4
AN=KN
4y-3=9→4y=9+3
4y=12
Y=3

8. Practice Example
Solve for x
4+x=2x-12
X=16
2x=x+3
X=3

9. Example#1
Solve for x.
Solution
5x-4=x+6
X=10/4

10. Example#2
Solve for x
Find the circumcenter of DEFG with E(4, 4), F(4, 2), and G(8, 2).
(6,3)
Need to do the distance formula for each line,
√(0-0)²+(0-8)²= 8
√(0-10)²+(0-8)²=2√41
√(0-10)²+(8-8)²=10

11. Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle intersect at a point called the incenter of the triangle, which is equidistant from the sides of the triangle.
Point of concurrency of the angle bisectors of a triangle

12. Practice Example Find the value of x?
The angle bisectors intersect at P.
The incenter P is equidistant from the sides, so SP = PT.
Therefore, x = 9.
Note that , the continuation of the angle bisector, is not the correct segment to use for the shortest distance from P to .

13. Examples
Find the value of x?
X=14
X=16
X=2

14. Median of a triangles
Median: A segment from a vertex to the midpoint of the opposite side

15. Centroids
The medians of ABC are AM BL,and CX.
The point of concurrency of the medians is called the centroid.
The centroid is point D.

16. Altitude of a triangle
Altitude: The perpendicular segment from a vertex to the line that contains the opposite side.
In an acute triangle all of the altitudes are all inside the triangle.

17. Altitudes of Right and Obtuse Triangle
In a right triangle, two of the altitudes are the legs of the triangle and the third is inside the triangle.
In an obtuse triangle, two of the altitudes are outside of the triangle.

18. Orthocenter
The orthocenter is the point of concurrency for the altitudes.
The altitudes of ∆QRS are AM, BL, and CX.
The orthocenter is point V.

19. Examples Determine whether is a median, an altitude, or neither.

20. Example Find the coordinates of the orthocenter of ∆ABC.

21. Real World Connections

22. Summary Meet at a single point
Perpendicular Bisectors in Triangles
Angle Bisector in Triangles
Meet at a single point
That is equidistant from the vertices of triangle
Point of concurrency is circumcenter of the triangle
Meet at a single point
that is equidistant from the sides of triangle
Point is incenter of an inscribed circle
Circle is inscribed in triangle

23. Ticket Out and Homework
Sect. 5-3
pg. 319 #'s 4,7,11
Pg. 322 #'s 4-8,10
Sect. 5-4
Pg327-8 #’s 7- 10,14,15
Pg331-2#’s 7-9,12
What is true of the segments that connect the incenter and circumcenter to each side of the triangle?