1 bisectors, medians, and altitudes section 5-1 agenda: 11/30/11 do now problem involving isosceles...
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Bisectors, Medians, and Altitudes
Section 5-1
Agenda:11/30/11
Do Now•Problem involving Isosceles and Equilateral TrianglesReview Do Now
Vocabulary: Perpendicular bisector, angle bisector, distance from a point to a line
Mini Lesson:•Using properties of Perpendicular bisectors and Angle Bisectors to solve problems
Independent /Group work•Practice problems
Share-out•Discussion of answers
Wrap-Up/ Summary•Writing Exercise
Lesson Quiz
Homework•Review Class Notes, Castle Learning
Mrs. PadillaGeometry Fall 2011
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• To identify and use perpendicular
bisectors & angle bisectors in triangles
• To identify and use medians & altitudes in triangles
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• Perpendicular Bisectors• Angle Bisectors• Locus• Equidistant• Medians• Altitudes• Points of Concurrency
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
Perpendicular bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.
Theorem 5-1-: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.
Theorem 5-2-: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.
• If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
If C is on the perpendicular bisector of AB, then CA = CB. ~
AM B
CIF
AM B
C
THEN
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of a segment.
A
C
B
D
If DA DB,
then D lies on
the perpendicular
bisector of AB.
P
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• For every triangle there are 3 perpendicular bisectors• The 3 perpendicular bisectors intersect in a common point named the circumcenter.
In the picture to the rightpoint K is the circumcenter.
Angle bisector of a triangle: A segment that bisects an angle of a triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle.
Theorem 5-3: Any point on the bisector of an angle is equidistant from the sides of the angle.
Theorem 5-4: 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.
• If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.IF THEN
• If m< 1 = m< 2, then BC = BD.
A
B1
2A
B1
2
C
D
~
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• For every triangle there are 3 angle bisectors.• The 3 angle bisectors intersect in a common point
named the incenter
In the picture to the right, point I is the incenter.
Median: A segment that connects a vertex of a triangle to the midpoint of the side opposite to that vertex. Every triangle has three medians.
Altitude: A segment that has an endpoint ata vertex of a triangle and the other on the lineopposite to that vertex, so that the segment is perpendicular to this line. Do example 1, page 239
Altitudes ofa right triangle
Altitudes ofan obtusetriangle
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A line segment whose endpoints are a vertex of atriangle and the midpoint of the side opposite thevertex.
In the picture to the right, the blue line segment is the median.
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• For every triangle there are 3 medians• The 3 medians intersect in a common point named the
centroid
In the picture to the right, point O is the centroid.
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A line segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
In the picture above, ∆ABC is an obtuse triangle & ∠ACB is the obtuse angle. BH is an altitude.
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• For every triangle there are 3 altitudes• The 3 altitudes intersect in a common point called the orthocenter.
In the picture to the right, point H is the orthocenter.
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Concurrent Lines3 or more lines that intersect at a common point
Point of ConcurrencyThe point of intersection when 3 or more lines intersect.
Type of Line Segments Point of ConcurrencyPerpendicular Bisectors CircumcenterAngle Bisectors IncenterMedian CentroidAltitude Orthocenter
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Facts to remember:1. The circumcenter of a triangle is equidistant from the
vertices of the triangle.2. Any point on the angle bisector is equidistant from the
sides of the angle (Converse of #3)3. Any point equidistant from the sides of an angle lies on
the angle bisector. (Converse of #2)4. The incenter of a triangle is equidistant from each side
of the triangle.5. The distance from a vertex of a triangle to the centroid
is 2/3 of the median’s entire length. The length from the centroid to the midpoint is 1/3 of the length of the median.
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• Use the diagram to find AB.
In the diagram, AC is the perpendicular bisector of DB. Therefore AB = AD
8x = 5x + 123x = 12
x = 4Since you were asked for AB, not just x:AB = 8x = 8 • 4 = 32
A
BC
D
8x5x + 12
Example
Is a perpendicular bisector of ? Why, or why not?CD
AB
A
C
B
D
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
ExamplesDoes the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?
A
A
B
BC
C
A
B
CP
P
D
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
ExamplesDoes the information given in the diagram allow you to conclude that P is on the angle bisector of angle A?
P
6
P
P6
A A
A
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
How can you tell if a ray or line segment is an angle bisector?
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
How can you tell if a ray or line segment is a perpendicular bisector?
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1. Perpendicular Bisectors
2. Angle Bisectors
3. Medians
4. Altitudes
1. …form right angles AND 2 lines segments
2. …form 2 angles
3. …form 2 line segments
4. … form right angles
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
1. Does D lie on the perpendicular bisector of
? ?WhyABA
C
B
D67
Draw the diagram and answer the question
Review Class Notes
Sec 5.1N and Sec 4.8R
AIM: How do we use properties of Perpendicular bisectors and Angle Bisectors ?
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(Finally!)Oh yeah! Do homework tonight and STUDY these notes
that you just took on Section 5-1!