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!