### Chapter 5: Relationships in Triangles

```MODELING MONDAY
RECAP
Take the last power of 2 that occurs before the number
of seats.
Take the number of seats minus that power of 2.
This is your seat you should pick!
Chapter 5: Relationships in Triangles
• New Homework Calendar
• Chapter 5 test: December 19th
5-1 BISECTORS OF
TRIANGLES
Objective: Identify and use perpendicular
bisectors and angle bisectors in triangles.
Perpendicular Bisectors
Example 1
A. Find BC.
Use the Perpendicular Bisector Theorems
Example 1
B. Find XY.
Use the Perpendicular Bisector Theorems
Example 1
C. Find PQ.
Use the Perpendicular Bisector Theorems
Try with a partner
A. Find NO.
A. 4.6
B. 9.2
C. 18.4
D. 36.8
TOO
B. Find TU.
A. 2
B. 4
C. 8
D. 16
TOO
C. Find EH.
A. 8
B. 12
C. 16
D. 20
Circumcenter Theorem
Example 2 (just watch.. Don’t write)
GARDEN A triangular-shaped garden is shown.
Can a fountain be placed at the circumcenter and
still be inside the garden?
By the Circumcenter Theorem, a point equidistant from
three points is found by using the perpendicular bisectors
of the triangle formed by those points.
Use the Circumcenter Theorem
Example 2 (continued)
Copy ΔXYZ, and use a ruler and protractor to draw the
perpendicular bisectors. The location for the fountain is
C, the circumcenter of ΔXYZ, which lies in the exterior of
the triangle.
C
Answer: No, the circumcenter of an obtuse triangle is in
the exterior of the triangle.
Use the Circumcenter Theorem
Think-Pair-Share
BILLIARDS A triangle used to
rack pool balls is shown. Would
the circumcenter be found
inside the triangle?
A. No, the circumcenter of an acute
triangle is found in the exterior
of the triangle.
B. Yes, circumcenter of an acute
triangle is found in the interior of
the triangle.
Angle Bisectors
Example 3
A. Find DB.
Use the Angle Bisector Theorems
Example 3
B. Find m Ð WYZ.
Use the Angle Bisector Theorems
Example 3
C. Find QS.
Answer: So, QS = 4(3) – 1 or 11.
Use the Angle Bisector Theorems
A. Find the measure of SR.
A. 22
B. 5.5
C. 11
D. 2.25
Example 3
B. Find the measure of <HFI.
A. 28
B. 30
C. 15
D. 30
Example 3
C. Find the measure of UV.
A. 7
B. 14
C. 19
D. 25
Incenter Theorem
Example 4
A. Find ST if S is the incenter
of ΔMNP.
By the Incenter Theorem, since S is
equidistant from the sides of ΔMNP,
ST = SU.
Find ST by using the
Pythagorean Theorem.
a2 + b2 = c2
Pythagorean Theorem
82 + SU2 = 102
Substitution
64 + SU2 = 100
82 = 64, 102 = 100
Use the Incenter Theorem
Example 4
SU2 = 36
SU = ±6
Subtract 64 from
each side.
Take the square root
of each side.
Since length cannot be negative, use only the positive
square root, 6. Since ST = SU, ST = 6.
Use the Incenter Theorem
Example 4
B. Find m<SPU if S is the incenter
of ΔMNP.
1 (62) or 31
2
Use the Incenter Theorem
Try with a Partner
A. Find the measure of GF if D is the
incenter of ΔACF.
A. 12
B. 144
C. 8
D. 65
TOO
B. Find the measure of <BCD if D is
the incenter of ΔACF.
A. 58°
B. 116°
C. 52°
D. 26°
Homework
```