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04/19/23 5-1: Special Segments in Triangles
Perpendicular Bisectors of a TriangleDefn: 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.
04/19/23 5-1: Special Segments in Triangles
Perpendicular Bisectors of a TriangleEvery triangle has 3 perpendicular bisectors.
04/19/23 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.
04/19/23 5-1: Special Segments in Triangles
Angle Bisectors of TrianglesDefn: 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.
04/19/23 5-1: Special Segments in Triangles
Angle Bisectors of TrianglesEvery 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.
04/19/23 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
T
S
U
04/19/23 5-1: Special Segments in Triangles
Angle Bisector Theorem
If D is on the bisector of ∠ABC, then
A
Y
X
D
C
B
DX = DY.
04/19/23 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.
04/19/23 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.
Finding the Centroid of a Triangle
Find the coordinates of the centroid of JKL.
SOLUTION (7, 10)
(3, 6)
(5, 2)
J
K
L
N
P
04/19/23 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.
04/19/23 5-1: Special Segments in Triangles
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.
04/19/23 5-1: Special Segments in Triangles
Altitudes of Triangles
Every triangle has 3 altitudes that will always intersect in the same point.
04/19/23 5-1: Special Segments in Triangles
Altitudes of Triangles
If the triangle is acute, then the altitudes are all in the interior of the triangle.
04/19/23 5-1: Special Segments in Triangles
Altitudes of TrianglesIf the triangle is a right triangle, then one altitude is in the interior and the other 2 altitudes are the legs of the triangle.
04/19/23 5-1: Special Segments in Triangles
Altitudes of TrianglesIf 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.
04/19/23 5-1: Special Segments in Triangles
ZC is an altitude, m CYW = 9x + 38 and ∠m WZC = 17x. Find m WZC.∠ ∠
Y
X
Z
W
A
C