1. Perpendicular Bisectors of a Triangle
Defn: Perpendicular Bisector of a Triangle: A segment is a perpendicular bisector of a triangle iff it is the perpendicular bisector of a side of the triangle.
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2. Perpendicular Bisectors of a Triangle
Every triangle has 3 perpendicular bisectors.
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3. Perpendicular Bisectors of a Triangle
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4. Perpendicular Bisectors of a Triangle
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5. Perpendicular Bisectors of a Triangle
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6. 5-1: Special Segments in Triangles
The 3 perpendicular bisectors of any triangle will intersect at a point that is equidistant from the vertices of the triangle.
This point is called the circumcenter and is the center of a circle that contains all 3 vertices of the triangle.
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7. Angle Bisectors of Triangles
Defn: Angle Bisector of a Triangle: A segment is an angle bisector of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is any other point on the triangle such that the segment bisects an angle of the triangle.
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8. Angle Bisectors of Triangles
Every triangle has 3 angle bisectors which will always intersect in the same point - the incenter. The incenter is the same distance from all 3 sides of the triangle. The incenter of a triangle is also the center of a circle that will intersect each side of the triangle in exactly one point.
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9. Angle Bisectors of Triangles
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10. Angle Bisectors of Triangles
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11. Angle Bisectors of Triangles
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12. 5-1: Special Segments in Triangles
RU is an angle bisector, m∠RTU = 13x – 24, m∠TRS = 12x – 34 and m∠RUS = 92. Determine m∠RSU. Is RU ⊥ TS? R S T U
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13. Angle Bisector Theorem
If D is on the bisector of ∠ABC, then X D B Y C
DX = DY.
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14. 5-1: Special Segments in Triangles
Median of a Triangle
Defn: Median of a Triangle: A segment is a median of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is the midpoint of the side opposite that vertex.
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15. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
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16. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
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17. 5-1: Special Segments in Triangles
Medians of a Triangle
Every triangle has 3 medians.
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18. 5-1: Special Segments in Triangles
Centroids
The medians of a triangle will always intersect at the same point - the centroid. The centroid of a triangle is located 2/3 of the distance from the vertex to the midpoint of the opposite side.
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19. Average of the three vertices
Formula for finding the centroid of a triangle
Average of the three vertices

20. (7, 10) (3, 6) (5, 2) Find the coordinates of the centroid of  JKL.
Finding the Centroid of a Triangle
Find the coordinates of the centroid of  JKL.
SOLUTION J K L N P
(7, 10)
(3, 6)
(5, 2)

21. 5-1: Special Segments in Triangles
Centroid
centroid
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22. 5-1: Special Segments in Triangles
Centroid
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23. 5-1: Special Segments in Triangles
Points U, V, and W are the midpoints of YZ, XZ and XY respectively. Find a, b, and c.
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24. Altitudes of Triangles
Defn: Altitude of a Triangle: A segment is an altitude of a triangle iff one endpoint is a vertex of the triangle and the other endpoint is on the line containing the opposite side such that the segment is perpendicular to line.
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25. Altitudes of Triangles
Every triangle has 3 altitudes that will always intersect in the same point.
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26. Altitudes of Triangles
If the triangle is acute, then the altitudes are all in the interior of the triangle.
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27. Altitudes of Triangles
If the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle.
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28. Altitudes of Triangles
If the triangle is an obtuse triangle, then one altitude is in the interior and the other 2 altitudes are in the exterior of the triangle.
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29. Altitudes of Triangles
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30. ZC is an altitude, m∠CYW = 9x + 38 and m∠WZC = 17x. Find m∠WZC.
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31. 5-1: Special Segments in Triangles
In ΔABC below, AB ≅ BC and AD bisects ∠BAC. If the length of BD is 3(x + 2) units and BC = 42 units, what is the value of x? 5 6 12 13 A C D B
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