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.
Timer
Do Now: Throwback! Using the compass and straightedge, construct a perpendicular bisector.
Name:
Date: January 12, 2016
Unit: Relationships Within Triangles
Topic: Perpendicular Bisectors
Aim: How can we identify properties of perpendicular bisectors?


Timer
Homework: Worksheet Due Monday 1/19/16. Benchmark 1/19/16
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
Obtuse triangle
Timer
Do Now: Throwback: Bisect an acute angle and an obtuse angle.
Name:
Date: January 13, 2016
Unit: Relationships Within Triangles
Topic: Angle Bisectors
Aim: How can we identify properties of angle bisectors?


Timer
Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16
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
Construct the incenter of:
An acute triangle
A right triangle
An obtuse triangle
Find the value of x




Bonus: Find the circumcenter of ∆ABC:
A(5,2), B(-1,2), C(-1,-3)
A(2,-2), B(-4,-2), C(-4, -7)
Timer
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?
Name:
Date: January 14, 2016
Unit: Relationships Within Triangles
Topic: Medians and Altitudes
Aim: How can we identify properties of medians and altitudes of a triangle?


Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16
Timer
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?
Example 2
For ∆PQS, is PR a median, altitude or neither? Explain
Is QT a median, altitude or neither? Explain
Independent Practice
Timer
Do Now: Algebra Throwback!
Solve the following inequalities:
Name:
Date: January 15, 2016
Unit: Relationships Within Triangles
Topic: inequalities in one triangle
Aim: How can we use inequalities involving angles and sides of triangles?


Homework: Worksheet Due Tuesday 1/19/16. Benchmark 1/20/16
Timer
Triangle Inequalities
Example 1
A town park is triangular. A landscape architect wants to place a bench at the corner with the largest angle. Which two streets form the corner with the largest angle?
Example 2
List the sides of ∆TUV in order from shortest to longest.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Example:
Example 3
Can a triangle have sides with the given lengths? Explain.
3 ft, 7 ft, 8 ft


5 ft, 10 ft, 15 ft
Example 4
Two sides of a triangle are 5 ft and 8 ft long. What is the range of possible lengths for the third side?
Independent Practice
Timer
Do Now: Algebra Throwback!
Solve the following equation using PEMDAS

-3 * ( 5x + 8 ) - 22 / 4 + 3x
Name:
Date: January 22, 2016
Unit: Relationships Within Triangles
Topic: points of Concurrency
Aim: How can we review points of concurrency?


Homework: Come up with your own way to remember points of concurrency
Timer
How to remember Points of Concurrency
All Of : Altitudes - Orthocenter
My Children: Medians - Centroid
Are Bringing In: Angle Bisectors - Incenter
Peanut Butter Cookies: Perpendicular Bisectors - Circumcenter
Do Now: In the diagram, the perpendicular bisectors (shown with dashed segments) of MNP meet at point O—the circumcenter. Find the indicated measure.
Name:
Date: January 25, 2016
Unit: Relationships Within Triangles
Topic: points of Concurrency
Aim: How can we review points of concurrency?


Homework: Pass your regents
Timer
1. MO = ___________ 2. PR = __________
 
3. MN = __________ 4. SP = __________
 
5. mMQO = __________
 
6. If OP = 2x, find x.
Example
Point S is the centroid of RTW, RS = 4, VW = 6, and TV= 9. Find the length of each segment.
RV = __________
 
SU = __________
 
RU = __________
 
RW = __________
 
TS = __________
 
SV = __________