1. Math Warm-ups
1-8-09
Fire stations are located at A and B. XY , which contains
Havens Road, represents the perpendicular bisector of AB .
A fire is reported at point X. Which fire station is
closer to the fire? Explain.

2. What is the incenter of a triangle?
Math Warm-ups
1-9-09
What is the incenter of a triangle?

3. A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Every triangle has three medians, and the medians are concurrent.

4. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.

5. Math Support Warm-ups
1-8-09
True or false:
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.

6. Vocabulary concurrent point of concurrency circumcenter of a triangle
incenter of a triangle
Orthocenter
Centroid

7. Point of concurrency
When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is the point where they intersect.

8. Incenter of a circle The incenter is always inside the triangle.
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .
The incenter is always inside the triangle.

9. Incenter of a circle The incenter is always inside the triangle.
A triangle has three angles, so it has three angle bisectors. The angle bisectors of a triangle are also concurrent. This point of concurrency is the incenter of the triangle .
The incenter is always inside the triangle.

10. Incenter of a circle
The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
Point of concurrency is center of imaginary inscribed circle
Incenter is equidistant from the sides
PX = PY = PZ

11. Using Properties of Angle Bisectors
Incenter Example:
Using Properties of Angle Bisectors
MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.
P is the incenter of ∆LMN. By the Incenter Theorem, P is equidistant from the sides of ∆LMN.
The distance from P to LM is 5. So the distance from P to MN is also 5.

12. Circumcenter center of a circle
The three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.
The circumcenter can be inside the triangle, outside the triangle, or on the triangle.

13. Point of concurrency is center of circumscribed circle
Circumcenter of a triangle
The three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.
The circumcenter can be inside, outside, or on the triangle.
Point of concurrency is center of circumscribed circle
Circumcenter is equidistant from the vertices
PA = PB = PC

14. Using Properties of Perpendicular Bisectors
Circumcenter Example:
Using Properties of Perpendicular Bisectors
DG, EG, and FG are the perpendicular bisectors of ∆ABC. Find GC.
G is the circumcenter of ∆ABC. By the Circumcenter Theorem, G is equidistant from the vertices of
∆ABC.
GC = GB
Circumcenter Thm.
GC = 13.4
Substitute 13.4 for GB.

15. Centroid of a Triangle
The point of concurrency of the median of a triangle is the centroid of the triangle . The centroid is always inside the triangle. The centroid is also called the center of gravity because it is the point where a triangular region will balance.
AP to PY is 2:1
CP to PX is 2:1
ZP to PB is 2:1

16. Using the Centroid to Find Segment Lengths
Centroid Example
Using the Centroid to Find Segment Lengths
In ∆LMN, RL = 21 and SQ =4. Find LS.
Centroid Thm.
Substitute 21 for RL.
LS = 14
Simplify.

17. The Orthocenter of a Circle
The point of concurrency of the altitudes of a triangle is the orthocenter of the triangle.

18. The Orthocenter of a Circle
In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.