Chương III. §5. Tính chất tia phân giác của một góc

Unit 5
ANGLE BISECTOR PROPERTIES


Angle Bisector Theorem


If a point is on the bisector of an angle, then the point is equidistant
from the sides of the angle.


Converse of the Angle Bisector Theorem


If a point in the interior of an angle is equidistant from the sides of the
angle, then the point is on the angle bisector.


Example 1


What is the length of line RM?


Example 2


What is the length of FB?


Independent Practice

1.

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Aim: How can we identify properties of perpendicular
bisectors?

Name:
Date: January 12, 2016
Unit: Relationships Within Triangles
Topic: Perpendicular Bisectors

Homework: Worksheet Due Monday 1/19/16. Benchmark 1/19/16

Do Now: Throwback! Using the compass and straightedge, construct a perpendicular bisector.

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Vocabulary


Concurrent – three or more lines intersect at one point



Point of concurrency – the point at which concurrent lines intersect



Circumcenter – point of concurrency of perpendicular bisectors



Circumscribed – when a circle surrounds another shape by touching
all the vertices of the shape.


Concurrency of perpendicular Bisectors
theorem


The perpendicular bisectors of the sides of a triangle are concurrent
at a point equidistant from the vertices


How to construct perpendicular Bisector point of
CONCURRENCY


Draw ∆ABC



Construct a perpendicular bisector of line AB



Construct a perpendicular bisector of line BC



Construct a perpendicular bisector of line AC



Label the point of intersection as P.


Concurrency of perpendicular Bisectors
theorem


The circumcenter of a triangle can be inside, on or outside a triangle.


Example 1


What are the coordinates of the circumcenter of the triangle with
vertices P(0,6), O(0,0), and S(4,0)?


Example 2


A town planner wants to locate a new fire station equidistant from the
elementary, middle and high schools. Where should she locate the

station?


Independent Practice


Construct perpendicular bisector concurrencies of a:

 Acute triangle
 Right triangle
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 Obtuse triangle


Aim: How can we identify properties of angle bisectors?
Name:
Date: January 13, 2016
Unit: Relationships Within Triangles
Topic: Angle Bisectors
Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16

Do Now: Throwback: Bisect an acute angle and an obtuse angle.

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Vocabulary


Incenter – point of concurrency of angle bisectors of a triangle



Inscribed – when the largest possible circle is inside a shape.


Concurrency of Angle Bisectors Theorem


The bisectors of the angles of a triangle are concurrent at a point
equidistant from the sides of the triangle.


Example 1


GE = 2x – 7 and GF = x + 4. What is GD?


Example 2


Name the point of concurrency of the angle bisectors


Independent Practice
1.

Construct the incenter of:

 An acute triangle
 A right triangle
 An obtuse triangle
Find the value of x

3.

Bonus: Find the circumcenter of ∆ABC:

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2.

1.
2.

A(5,2), B(-1,2), C(-1,-3)
A(2,-2), B(-4,-2), C(-4, -7)


Aim: How can we identify properties of medians and
altitudes of a triangle?

Name:
Date: January 14, 2016
Unit: Relationships Within Triangles
Topic: Medians and Altitudes

Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16

Do Now: Town officials want to place a recycling bin so that it is equidistant from the lifeguard chair, the
snack bar and the volleyball court. Where should they place it?

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Vocabulary


Median of a triangle – a segment whose
endpoints are a vertex and a midpoint of the
opposite side



Altitude of a triangle – the perpendicular
segment from a vertex of a triangle to the line
containing the opposite side.


Concurrency of Medians Theorem


The medians of a triangle are concurrent at a point that is two thirds
the distance from each vertex to the midpoint of the opposite side.
The point where the lines meet is called the centroid of the triangle.

Concurrency of Altitudes Theorem


The lines that contain the altitudes of a triangle are concurrent. The
point where the three altitudes meet is called the orthocenter of the
triangle. The orthocenter could be inside, on or outside the triangle.


Example 1


In the diagram below, XA = 8. What is the length of XB?