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For the following problems, use A(5,10), B(2,10), C(3,3) Find AB Find the midpoint of CA Find the midpoint of AB Find the slope of AB

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For the following problems, use A(5,10), B(2,10), C(3,3) Find AB

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For the following problems, use A(5,10), B(2,10), C(3,3) Find the midpoint of CA

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For the following problems, use A(5,10), B(2,10), C(3,3) Find the midpoint of AB

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For the following problems, use A(5,10), B(2,10), C(3,3) Find the slope of AB

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Section 5.2 Use Angle Bisectors of Triangles Section 5.3 Use Medians and Altitudes Section 5.4

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The Concept Yesterday we investigated the concept of circumcenters Today we’re going to look at a couple of other kinds of points of concurrency We’ll look at each of the theorems that helps us to create these points, while also enhancing our understanding of triangles

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What is the hallmark of an Isosceles Triangle? A Isosceles triangles can also be made by combining two right triangles A This creates a situation in which the base is twice the size of the previous triangle, as well the two end vertices are equidistant to the top vertex

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This isosceles triangle is also created when a perpendicular bisector extends from a point to a line segment AA This gives us the Perpendicular Bisector Theorem BB

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Theorem 5.2: Perpendicular Bisector Theorem In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. C B Theorem 5.3: Converse of the Perpendicular Bisector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment A

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Find the length of segment AB 5x C D B A 4x+3

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Theorem 5.4: Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle This point is called the circumcenter which means all circles created from it will include all of the vertices

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Theorem 5.5: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. C B Theorem 5.6: Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the sides of an angle, then it lies on the bisector of the angle A D

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Find x C B A D x 15 27 o

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Find x & y C B A D 5y-8 3y+12 35 o 4x-1 o

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Theorem 5.7: Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle This point of concurrency is called the incenter. Circles centered at this point will equally touch all three sides of the triangle

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5.2 Exercises 1-9, 16, 17, 20-25 5.3 Exercises 3-20, 24, 25

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Theorem 5.8: Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side Median: Segment from the vertex of a triangle to the midpoint of the opposite side This point of concurrency is called the centroid. The centroid is the center of area of the object. It’s the point on which we could balance the entire object

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Find x 10 x

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Find x & y if EB=15 and DA=14 A 2x-1 B C D E F G y+2 3y

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Altitude: Perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side

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Theorem 5.9: Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent Orthocenter

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5.4 Exercises 1-7, 12-15, 25-28, 33-35,39

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Perpendicular Bisector TheoremPerpendicular Bisector Theorem CircumcentersCircumcenters Angle Bisector TheoremAngle Bisector Theorem IncentersIncenters MediansMedians CentroidsCentroids AltitudesAltitudes OrthocentersOrthocenters

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