1. Geometry
Bisector of Triangles
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2. Warm up
Determine whether each point is on the perpendicular bisector of the segment with endpoints S(0,8) and T(4,0).
1) X(0,3) ) Y(-4,1) ) Z(-8,-2)
yes no
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3. Bisector of Triangles
Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property.
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4. Draw a large scalene acute triangle ABC on a piece of patty paper. 1 2 3 B A B P C A C A C
Draw a large scalene acute triangle ABC on a piece of patty paper.
Fold the perpendicular bisector of each side.
Label the point where the three perpendicular bisectors intersect as P.
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5. 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. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.
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6. Circumcenter Theorem Theorem 2.1 B
The circumcenter of a triangle is equidistant from the vertices of the triangle.
PA = PB = PC P A B C
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7. The circumcenter can be inside the triangle, Out the triangle, or on the triangle.P
Acute triangle
Obtuse triangle
Right triangle
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8. The circumcenter of ∆ABC is the center of its circumscribed circle
The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. B P A C
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9. Circumcenter Theorem A l n P B C m
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10. Using Properties of Perpendicular BisectorH J G K L
19.9 Z M
18.6
14.5
9.5
Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ.
HZ = GZ Circumcenter Thm.
HZ = Substitute 19.9 for GZ.
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11. Use the diagram. Find each length.
Now you try!
Use the diagram. Find each length.
a) GM b) GK c) JZ H J G K L
19.9 Z M
18.6
14.5
9.5
1a) 14.5
1b) 18.6
1c) 19.9
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12. Finding the Circumcenter of a Triangle.
Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0).
Step 1 Graph the triangle. y
X = -4 S
Y = 2
(-3,2) x R 4 O
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13. Step 2 Find equation for two perpendicular bisectors. Since
Two sides of the triangle lie along the axes, use the graph to
find the perpendicular bisectors of these two sides. The
perpendicular bisector of RO is x = -3, and the perpendicular
bisector of OS is y =2. y
X = -4 S
Y = 2
(-3,2) x R 4 O
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14. Step 3 Find the intersection of the two equations.
The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y
X = -4 S
Y = 2
(-3,2) x R 4 O
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15. 2) Find the circumcenter of ∆GOH with vertices
Now you try!
2) Find the circumcenter of ∆GOH with vertices
G(0,-9), O(0,0), and H(8,0).
2) 29
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16. Theorem 2.2 Incenter Theorem
The incenter of a triangle is equidistant from the sides of the triangle.
PX = PY = PZ B Z Y P C A X
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17. Unlike the circumcenter, the incenter is always inside the triangle.P P P
Acute triangle
Obtuse triangle
Right triangle
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18. The incenter is the center of the triangle’s inscribed 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. P A B C
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19. Using Properties of Angle BisectorsK
A) The distance from V to KL
7.3
V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL.
The distance from V to JK is 7.3.
So the distance from V to KL is also 7.3 W V
106˚ J L
19˚
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20. JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K
B) m/ Vkl
JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K
m/ kJL = 2m / VJL
m/ kJL = 2(19˚) = 38˚
m/ kJL + m/ JLK + m/ JKL = 180˚
m/ JKL = 180˚
m/ JKL = 36˚
m/ Vkl = ½ m/ JKL
m/ Vkl = ½ (36˚) = 18˚
Substitute the given values. Subtract 144˚ from both sides.
7.3 W
KV is the bisector of m/ JKL V
Substitute 36˚ for m/ JKL.
106˚ L
19˚
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21. Now you try! 3) QX and RX are angle bisectors ∆PQR. Find each measure.
The distance from X to PQ
m/ PQX Q X
3a) 14.5
3b) 18.6 R P
12˚ Y
19.2
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22. Community Application
The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A
Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B
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23. Trace the map. Draw the triangle formed by the viewing locations
Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A B C F
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24. Now you try!
4) A City plans to build a firefighters’ monument in the park between three streets. Draw a sketch to show where the city should place the monument so that it is the same distance from all three streets. Justify your sketch.
Centerville Anverue
King Boulevard
Third Street
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25. Now some practice problems for you!
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26. Assessment
SN, TN, and VN are the perpendicular bisectors of ∆PQR. Find each length.
1) NR ) RV
3) TR ) QN Q
3.95 S N T
5.64
4.03 P R
5.47 V
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27. Find the circumcenter of a triangle with the given vertices.
5) O(0,0), K(0, 12), L(4,0)
6) A(-7,0), O(0,0),B(0,-10)
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28. CF and EF are angle bisectors of ∆CDE. Find each measure.
7) The distance from F to CD
8) m/ FED C D E G F
42.1
54˚
17˚
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29. 9) The designer of the Newtown High School pennant wants the circle around the bear emblem to be as large as possible. Draw a sketch to show where the center of the circle should be located. Justify your sketch.
BEARS
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30. Let’s review Bisector of Triangles
Since a triangle has three sides, it has three perpendicular bisector. When you construct the perpendicular bisectors, you find that they have an interesting property.
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31. Draw a large scalene acute triangle ABC on a piece of patty paper. 1 2 3 B A B P C A C A C
Draw a large scalene acute triangle ABC on a piece of patty paper.
Fold the perpendicular bisector of each side.
Label the point where the three perpendicular bisectors intersect as P.
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32. 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. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the circumcenter of the triangle.
CONFIDENTIAL

33. Theorem 2.1 Circumcenter Theorem B
The circumcenter of a triangle is equidistant from the vertices of the triangle.
PA = PB = PC P C A
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34. The circumcenter can be inside the triangle, Out the triangle, or on the triangle.P P P
Acute triangle
Obtuse triangle
Right triangle
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35. The circumcenter of ∆ABC is the center of its circumscribed circle
The circumcenter of ∆ABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is circumscribed about the polygon. B P A C
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36. Circumcenter Theorem A l n P B C m
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37. Using Properties of Perpendicular BisectorH
18.6
Z is the circumcenter of ∆GHJ. By the Circumcenter Theorem, Z is equidistant from the vertices of ∆GHJ.
HZ = GZ Circumcenter Thm.
HZ = Substitute 19.9 for GZ. Z L K
9.5
19.9 G J M
14.5
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38. Finding the Circumcenter of a Triangle.
Finding the circumcenter of ∆RSO with vertices R(-6,0), S(0,4), and O (0,0).
Step 1 Graph the triangle. y
X = -4 S
Y = 2
(-3,2) x R 4 O
Next page:
CONFIDENTIAL

39. Step 2 Find equation for two perpendicular bisectors. Since
Two sides of the triangle lie along the axes, use the graph to
find the perpendicular bisectors of these two sides. The
perpendicular bisector of RO is x = -3, and the perpendicular
bisector of OS is y =2. y
X = -4 S
Y = 2
(-3,2) x R 4 O
Next page:
CONFIDENTIAL

40. Step 3 Find the intersection of the two equations.
The lines x = -3 and y = z intersect at (-3,2), the circumcenter of ∆RSO. y
X = -4 S
Y = 2
(-3,2) x R 4 O
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41. Theorem 2.2 Incenter Theorem
The incenter of a triangle is equidistant from the sides of the triangle.
PX = PY = PZ B Z Y P C A X
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42. Unlike the circumcenter, the incenter is always inside the triangle.P P P
Acute triangle
Obtuse triangle
Right triangle
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43. The incenter is the center of the triangle’s inscribed 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. B P A C
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44. Using Properties of Angle BisectorsK
A) The distance from V to KL
7.3
V is the incenter of ∆JKL. By the Incenter Theorem, V is equidistant from the sides of ∆JKL.
The distance from V to JK is 7.3.
So the distance from V to KL is also 7.3 W V
106˚ J L
19˚
Next page:
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45. JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K
B) m/ Vkl
JV is the bisector of m/ kJL. Substitute 19˚ for m / VJL. ∆ sum Thm. K
m/ kJL = 2m / VJL
m/ kJL = 2(19˚) = 38˚
m/ kJL + m/ JLK + m/ JKL = 180˚
m/ JKL = 180˚
m/ JKL = 36˚
m/ Vkl = ½ m/ JKL
m/ Vkl = ½ (36˚) = 18˚
Substitute the given values. Subtract 144˚ from both sides.
7.3 W
KV is the bisector of m/ JKL V
Substitute 36˚ for m/ JKL.
106˚ L
19˚
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46. Community Application
The city of Odessa will host a fireworks display for the next Fourth of July celebration. Draw a sketch to show where the display should be positioned so that it is the same distance from all three viewing location A, B, and C on the map. Justify your sketch. A
Let the three viewing locations be vertices of a triangle. By the Circumcenter Theorem, the circumcenter of the triangle is equidistant from the vertices. C B
Next page:
CONFIDENTIAL

47. Trace the map. Draw the triangle formed by the viewing locations
Trace the map. Draw the triangle formed by the viewing locations. To find the circumcenter, find the perpendicular bisectors of each side. The position of the display is the circumcenter, F. A C F B
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48. You did a great job today!
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