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Learn about the definitions and properties of special segments in triangles, including medians, altitudes, angle bisectors, and perpendicular bisectors.
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Lesson: 5.1 Special Segments in Triangles Pages: 238 – 241 Objectives: • To IDENTIFY and USE: MEDIANs ALTITUDEs ANGLE BISECTORs PERPENDICULAR BISECTORs in Triangles
GEOMETRY 5.1 Special Segments in Triangles DEFINITION: A line or line segment that passes through the MIDPOINT of a side of a Triangle and is PERPENDICULAR to that side is a PERPENDICULAR BISECTOR of the side of the Triangle
GEOMETRY 5.1 PERPENDICULAR BISECTOR POINTS Theorem Any POINT on the PERPENDICULAR BISECTOR of a Segment Is EQUIDISTANT from the ENDPOINTS of the Segment.
GEOMETRY 5.1 EQUIDISTANT ENDPOINTS Theorem A Line containing Two Points, each EQUIDISTANT from the ENDPOINTS of a given SEGMENT, Is the PERPENDICULAR BISECTOR of the Segment.
GEOMETRY 5.1 A Line containing Two Points, each EQUIDISTANT from the ENDPOINTS of a given SEGMENT, Is the PERPENDICULAR BISECTOR of the Segment.
GEOMETRY 5.1 A Line containing Two Points, each EQUIDISTANT from the ENDPOINTS of a given SEGMENT, Is the PERPENDICULAR BISECTOR of the Segment.
GEOMETRY 5.1 A Line containing Two Points, each EQUIDISTANT from the ENDPOINTS of a given SEGMENT, Is the PERPENDICULAR BISECTOR of the Segment. GIVEN: PROVE:
GEOMETRY 5.1 GIVEN: PROVE:
GEOMETRY 5.1 MEDIAN of a Triangle -- A segment that CONNECTS a vertex of the triangle and the midpoint of the opposite side.
GEOMETRY 5.1 Median
GEOMETRY 5.1 Median How many Medians are there in a Triangle?
GEOMETRY 5.1 How many Medians are there in a Triangle?
GEOMETRY 5.1 EVERY Triangle has THREE Medians, One from EACH Vertex How many Medians are there in a Triangle?
GEOMETRY 5.1 ALTITUDE of a Triangle -- A segment from a vertex of the triangle PERPENDICULAR to the linecontaining the OPPOSITE Side.
GEOMETRY 5.1 ALTITUDE
GEOMETRY 5.1 Every Triangle has 3 Altitudes, one for EACH Vertex. In the ACUTE Triangle, all Altitudes are INSIDE the Triangle.
BUT -- Altitudes do NOT have to be INSIDE the Triangle.
In this OBTUSE Triangle, 2 altitudes are OUTSIDE the Triangle. E B H A C G
In this Right Triangle, 2 Altitudes are the LEGS of the Triangle.
2 cm 1 cm 1 cm Where is the altitude and median of Triangle ABC?
Can an ALTITUDE also be a MEDIAN?
ISOSCELES Altitude Theorem The ALTITUDE from the Vertex Angle to the Base of an ISOSCELES Triangle Is a MEDIAN.
ISOSCELES Altitude Theorem The ALTITUDE from the Vertex Angle to the Base of an ISOSCELES Triangle Is a MEDIAN. This means: The Altitude of an Isosceles Triangle BISECTS the Base.
The ANGLE BISECTOR of a TRIANGE Bisects an ANGLE of a Triangle so that ONE END is at a VERTEX and the OTHER END is on the OPPOSITE SIDE.
The ANGLE BISECTOR of a TRIANGE Bisects an ANGLE of a Triangle so that ONE END is at a VERTEX and the OTHER END is on the OPPOSITE SIDE. Every Triangle has 3 Angle Bisectors.
ANGLE BISECTOR Theorem ANGLE BISECTOR Theorem Any POINT on the BISECTOR of an ANGLE is EQUIDISTANT from the SIDES of the ANGLE. Any POINT on the BISECTOR of an ANGLE is EQUIDISTANT from the SIDES of the ANGLE.
ANGLE BISECTOR Theorem Any POINT on the BISECTOR of an ANGLE is EQUIDISTANT from the SIDES of the ANGLE. Any POINT on or in the INTERIOR of an ANGLE and EQUIDISTANT from the SIDES of an ANGLE LIES on the BISECTOR of the ANGLE
2 3 1 In Triangle 1 -- draw a MEDIAN In Triangle 2 -- draw an ANGLE BISECTOR from Point C In Triangle 3 -- draw a PERPENDICULAR BISECTOR at Point C
Median, Angle Bisector, and Perpendicular are NOT necessarily the SAME UNLESS
Median, Angle Bisector, and Perpendicular are NOT necessarily the SAME UNLESS The Triangle is ISOSCELES!
ISOSCELES BISECTOR Corollary The BISECTOR of the Vertex of an Isosceles Triangle is also: a MEDIAN and an ALTITUDE of the Triangle.
Given: Prove:
