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Transcript of For the following problems, use A(5,10), B(2,10), C(3,3) Find AB Find the midpoint of CA Find the...
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BellworkFor 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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Bellwork SolutionFor the following problems, use A(5,10), B(2,10), C(3,3)•Find AB
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Bellwork SolutionFor the following problems, use A(5,10), B(2,10), C(3,3) Find the midpoint of CA
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Bellwork SolutionFor the following problems, use A(5,10), B(2,10), C(3,3)•Find the midpoint of AB
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Bellwork SolutionFor the following problems, use A(5,10), B(2,10), C(3,3)•Find the slope of AB
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Use Perpendicular BisectorsSection 5.2
Use Angle Bisectors of Triangles
Section 5.3
Use Medians and Altitudes
Section 5.4
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The ConceptYesterday we investigated the concept of circumcentersToday we’re going to look at a couple of other kinds of
points of concurrencyWe’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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Isosceles TrianglesWhat 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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Perpendicular BisectorsThis isosceles triangle is also created when a perpendicular
bisector extends from a point to a line segment
A A
This gives us the Perpendicular Bisector Theorem
B B
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TheoremsTheorem 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 TheoremIn 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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ExampleFind the length of segment AB
5x
CD
B
A
4x+3
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TheoremTheorem 5.4: Concurrency of Perpendicular Bisectors of a
TriangleThe 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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TheoremsTheorem 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 TheoremIf 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
AD
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ExampleFind x
C
B
AD
x
15
27o
27o
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ExampleFind x & y
C
B
AD
5y-8
3y+12
35o
4x-1o
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TheoremTheorem 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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Homework
5.2 Exercises1-9, 16, 17, 20-255.3 Exercises3-20, 24, 25
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Theorems
Theorem 5.8: Concurrency of Medians of a TriangleThe 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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ExampleFind x
10
x
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ExampleFind x & y if EB=15 and DA=14
A
2x-1
BC
D
E
F
G y+2
3y
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DefinitionAltitude:
Perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side
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TheoremsTheorem 5.9: Concurrency of Altitudes of a TriangleThe lines containing the altitudes of a triangle are concurrent
Orthocenter
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Homework
5.4 Exercises1-7, 12-15, 25-28, 33-35,39
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Most Important Points• Perpendicular Bisector Theorem• Circumcenters• Angle Bisector Theorem• Incenters• Medians• Centroids• Altitudes• Orthocenters