segments in triangles
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Segments in triangles. Triangle Geometry. Special Segments of a triangle: Median. Definition:. A segment from the vertex of the triangle to the midpoint of the opposite side. . Since there are three vertices in every triangle, there are always three medians . B. C. F. D. E. A. - PowerPoint PPT PresentationTRANSCRIPT
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SEGMENTS IN TR
IANGLES
T R I AN G L E G
E O M E T R Y
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2
SPECIAL SEGMENTS OF A TRIANGLE: MEDIANDefinition: A segment from the vertex of the triangle to the
midpoint of the opposite side. Since there are three vertices in every triangle, there are always three medians.
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WHERE THE MEDIANS MEET IN AN ACUTE TRIANGLE: THE CENTROID
B
A DE
CF
In the acute triangle ABD, figure C, E and F are the midpoints of the sides of the triangle. The point where all three medians meet is known as the “Centroid”. It is the center of gravity for the triangle.
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FINDING THE MEDIANS: AN ACUTE TRIANGLE
A
B
C
A
B
C
A
B
C
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FINDING THE MEDIANS: A RIGHT TRIANGLE
A
B
CA
B
C
A
B
C
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FINDING THE MEDIANS: AN OBTUSE TRIANGLE
A
B
C
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Special Segments of a triangle: Altitude
Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.
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ALTITUDES OF A RIGHT TRIANGLE
B
A D
F
In a right triangle, two of the altitudes of are the legs of the triangle.
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In a right triangle, two of the altitudes are legs of the right triangle. The third altitude is inside of the triangle.
ALTITUDES OF A RIGHT TRIANGLE
A
B
CA
B
C
A
B
C
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ALTITUDES OF AN OBTUSE TRIANGLE
In an obtuse triangle, two of the altitudes are outside of the triangle.
B
A D
F
I
K
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In an obtuse triangle, two of the altitudes are outside the triangle. For obtuse ABC:
BD is the altitude from BCE is the altitude from CAF is the altitude from A
ALTITUDE OF AN OBTUSE TRIANGLE
A
B
C
A
B
CA
B
CD
EF
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A
B
C
A
B
C
A
B
C
DRAW THE THREE ALTITUDES ON THE FOLLOWING TRIANGLE:
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DRAW THE THREE ALTITUDES ON THE FOLLOWING TRIANGLE:
A
B
C
A
B
C
A
B
C
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Draw the three altitudes on the following triangle:
A
B C
A
B C
A
B C
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SPECIAL SEGMENTS OF A TRIANGLE: PERPENDICULAR BISECTORThe perpendicular bisector of a segment is a line that is
perpendicular to the segment at its midpoint. The perpendicular bisector does NOT have to start at a vertex.
In the figure, line l is a perpendicular bisector of JK
J
K
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EXAMPLES:Draw the perpendicular bisector of the following lines, make
one a ray, one a line, and one a segment.J
K
A
B
X
Y
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Example:
C D
In the scalene ∆CDE, is the perpendicular bisector.
In the right ∆MLN, is the perpendicular bisector.
In the isosceles ∆POQ, is the perpendicular bisector.
EA
B
M
L N
A B
RO Q
P
FINDING THE PERPENDICULAR BISECTORS
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MIDSEGMENT THEOREM
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
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THRM 5-8 PG 177, THRM 5-9THRM 5-10 PG 178, THRM 5-11 (MIDSEGMENT THRM)
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PROOFS ON PG 177, 178,
Classroom exercises on pg 179
2 - 9
Written exercises on page 180
Extra examples
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PROOFS ON PG 177, 178,
Classroom exercises on pg 1792) 53) 144) 85) .5k6) A) 5 7 4, b 5 7 4c) 5 7 4, D 5 7 47) The segment joining the
midpoints of the sides of a triangle divide the triangle into 4 congruent triangles
8) 39) Thrm 5-8 (If two lines are
parallel then all points on one line are equidistant from the other line.
Written exercises on page 180
Extra examples