Transcript of By: Mariana Beltranena 9-5 PROPERTIES AND ATTRIBUTES OF TRIANGLES.
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- By: Mariana Beltranena 9-5 PROPERTIES AND ATTRIBUTES OF
TRIANGLES
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- Perpendicular Bisector The perpendicular bisector of a segment
is aligned perpendicular to other segment and to its midpoint.
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- Perpendicular Bisector Theorem If a point is in the
perpendicular bisector of a segment, then it is equidistant from
both endpoints.
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- and its Converse If a point is equidistant from both endpoints
of a segment then it is the perpendicular bisector of a
segment.
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- Angle Bisector Is a line that divides an angle into two
congruent angles.
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- Angle Bisector Theorem If a point is on the angle bisector of
an angle then the perpendicular distance to each side of the angle
is the same.
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- and its Converse If the perpendicular distance from a point to
both sides of an angle is the same, then the point is on the angle
bisector of that angle.
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- Concurrency of Perpendicular Bisectors Concurrent means the
coincidence on a point. Where three or more lines intersect at one
point.
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- The concurrency of bisectors is when the three perpendicular
bisectors of a triangle are on a point called circumcenter. This
point can be inside or outside of the triangle. Also around the
triangle it could be drawn a circle touching all of the
corners.
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- Circumcenter Therorem The circumcenter is equidistant from the
3 vertices of the triangle.
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- Concurrency of Angle Bisectors The three angle bisectors of a
triangle are concurrent on a point called incenter, which is always
in the triangle.
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- Incenter Theorem Is always in and is where the angle bisector
meets. The perpendicular distance from the incenter to the three
sides of the triangle is the same.
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- Medians The median of a side of a triangle is the line from the
midpoint of that side to the vertex opposite to it. Centroid: the
point of concurrency of the medians of a triangle. The centroid is
always inside of the triangle.
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- Centroid Theorem The distance from the vertex to the centroid
is 2/3 of the distance from the vertex to the opposite side
midpoint.
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- Concurrency of altitudes of a triangle theorem The three
altitudes of a triangle are concurrent in a point called
orthocenter which has no special properties. The altitude of a
triangle is the line from the vertex to its opposite side or the
prolongation of that side if the triangle is obtuse.
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- Midsegments The midsegment of a triangle is the line that joins
the midpoints of two of its sides. Midsegment Theorem The
midsegment of a triangle is parallel to the other side and measures
half the measure of that side.
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- Midsegment examples
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- the relationship between the longer and shorter sides of a
triangle and their opposite angles In the same triangle or in
congruent triangles with no congruent angles the side opposite to
the biggest angle is the biggest and the side opposite to the
smallest angle is the smallest side.
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- Triangle Inequality In a triangle inequality the sum of two
sides is always greater than the third side length.
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- Triangle inequality examples 1. Can 8,6,10 be the measures of a
triangle? If so tell if it is acute obtuse or right. 8+6>10
14>10 Yes 8+6=10 64+36=100 100=100 It is a right triangle
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- 2. Can 5,6,11 be a triangle? 2. 5+6 11.11= 11. No, because the
two short sides have to add up a greater number than the longer
side. 3. The measures of the sides of a triangle are 5 and 9. Find
all the possible measures of the third side. 2. 5+9= 14 3. 9-5=3 4.
x>3, x