1. Name the special segment Options: Median, Angle Bisector,
Warm Up:
Name the special segment
Options: Median, Angle Bisector,
Perpendicular Bisector, or Altitude

2. Points of Concurrency

3. Day 2:
Example:
CPR has vertices C(15, 1), P(9, 11), and R(2, 1). Determine the
coordinates of point A on CP so that RA is the median of CPR.  P R C

4. In PQR, QS and RT are medians. If PT=3x-1, PS= 4x - 2, and
SR = 2x + 4, find TQ.
What do we know about PS and SR?_______________________________
How does that help us?__________________________________ 
Once we find x, how can we find the measure of TQ?_________________________________

5. Example:
AM and CN are medians of  ABC, and LX is a median of  LMC. Find XM if BC=18.
How does MC relate to BC?_______________________________
How does XM relate to CM?_______________________________
How does that help us?__________________________________

6. In MOP, ∠O ≅ ∠MPO, m∠M = 40 degrees, and PN is an altitude.
Example:
In MOP, ∠O ≅ ∠MPO, m∠M = 40 degrees, and PN is an altitude.
Find m∠NPO.
What are the measures of <MPO and <O? ______________________________
What is the measure of <PNO?________________
<NPO=_____________

7. AC is an altitude of ABD, find m∠1.
Example:
AC is an altitude of ABD, find m∠1.
What is the measure of <ACB?____________________
How does that help us find <1?________________________
<1=________

8. In LMG, MK is an angle bisector, m∠1=2n+10, m∠2=4n−32 and m∠L=60.
Example:
In LMG, MK is an angle bisector, m∠1=2n+10, m∠2=4n−32 and
m∠L=60.
Find m∠G.
What do we know about <1 and <2?_______________________ How does that help us find
<G?______________________
<M = ______
<L  = ______
<M + <L + <G = _______ 
<G = _____

15. Warm Up: Name the point of concurrency where three medians intersect
Which two segments can intersect outside of a triangle?
What is an incenter?

16. Perpendicular Bisector Theorem:

17. PQ = 7 PQ = RQ Perpendicular Bisector Theorem
Example:
Find PQ.
PQ = RQ Perpendicular Bisector Theorem
3x + 1 = 5x – 3 Substitution
1 = 2x – 3 Subtract 3x from each side.
4 = 2x Add 3 to each side.
2 = x Divide each side by 2.
So, PQ = 3(2) + 1 = 7.
PQ = 7

18. Let’s remember what we know about Circumcenters…

19. EX: Given that O is the Circumcenter of Triangle ABC,
what do we know to be true?
OA= ?
OA= OC
OA= OB D E
AF= ?
AF= FB
AD= ?
AD= DC F
CE= ?
CE= EB

20. EX: Given that O is the Circumcenter of Triangle ABC, find
the measure of AO. 16 9
AO= 16

21. EX: CD is the perpendicular bisector of ABC.
Find the perimeter of triangle ABC.
8x-8
3x+2 3
PERIMETER:
AC+BC+AB
AB = 3+3
AB = 6
8+8+6 = 22
We know AC = BC
AC = 8x-8
AC = 8(2)-8
AC = 16-8
AC=8
and BC = 8
So…8x-8 = 3x+2
5x-8 = 2
5x = 10
x = 2

22. Angle Bisector Theorem:

23. QS = SR Angle Bisector Theorem 4x – 1 = 3x + 2 Substitution
Example: Find QS.
QS = SR Angle Bisector Theorem
4x – 1 = 3x + 2 Substitution
x – 1 = 2 Subtract 3x from each side.
x = 3 Add 1 to each side.
Answer: So, QS = 4(3) – 1 or 11.

24. Let’s remember what we know about Incenters…

25. D is the incenter of Triangle ACF. Find the length of BD.
BD = DG
BD = 5

26. Let’s remember what we know about Medians and Centroids…

27. In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.
Centroid Theorem
YV = 12
Simplify.
YP + PV = YV Segment Addition
8 + PV = 12 YP = 8
PV = 4 Subtract 8 from each side.
Answer: YP = 8; PV = 4

28. WP = (1/3) WZ 18 = (1/3) WZ 18 *3 = WZ 54 = WZ WZ = 54 PZ = 36
In ΔXYZ, P is the centroid and WP= 18. Find PZ and WZ.
WP = (1/3) WZ
18 = (1/3) WZ
18 *3 = WZ
54 = WZ
WZ = 54
PZ = 36
PZ = (2/3) WZ
PZ = (2/3) * 54
PZ = 36