Download 8-5 Angle Bisector Theorem

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Transcript
Name: ____________________________________
Geometry Per _____
8-5 Notes
Date: ___________
Angle Bisector Theorem
Learning Goals: (1) How can we use the angle bisector theorem to find missing segments in a given triangle?
Before we begin:
Factoring Practice
How to factor polynomials using sum and product:
How to factor and solve polynomials using sum and product:
Steps
The standard form for
a quadratic is
ax2 + bx + c = 0
1. We are looking for
factors of C
(numbers that
multiply to the
last number) that
also have the sum
of b (add to the
middle number)
2. List your factors of
c and identify
which pair add to b
3. Set up your
parenthesis to set
up your factors
appropriately.
Example
Together
ex 1) x2 + 5 x + 6 = 0
1. C = 6
Factors of 6:
1&6
2&3
-1 & -6
-2 & -3
2. Which of those
pairs add to 5?
2&3
3. (x+2)(x+3) = 0
x = -2
x = -3
Example Together
ex 2) -12 -x + x2 = 0
You Try!
ex 3) x2 + 7 x + 10 = 0
Together!
What type of segment is AP? How do you know?
With a partner!
 What do all three diagrams above have in common?
 Do you notice any relationships that exist?
ANGLE BISECTOR THEOREM: An _______________
of a vertex
angle in a triangle divides the ___________ side in two
segments that are _______________ to the
other two sides of the triangle.
LET’S TRY SOME:
̅̅̅̅ is the angle bisector of∠𝐵𝐴𝐶, 𝐴𝐷
̅̅̅̅divides the sides of the triangle proportionally.
1. Given that 𝐴𝐷
This proportion can be representd as,
*Is is true that AB * DC = AC * BD? Explain why.
2. Look at the measurements given below to determine if ̅̅̅̅
𝐴𝐻 is or is not an angle bisector. Explain your
reasoning.
Stations Practice - Work Space
Stations Practice - Work Space
Stations Practice
1. Find the missing
segment length in
the triangle below.
Explain how you
got your answer
(i.e., what theorem
did you use, etc…)
̅̅ an angle bisector? Justify
2. IS ̅̅
𝐶𝑆
your answer.
3. Given that ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹,
a)What is the ratio of their
corresponding medians?
b) What is the ratio of their areas?
4
a) Solve for x.
b) Using the answer
above, solve for CA.
5
Four streets in a town are
6.
illustrated in the accompanying
diagram. The angle between Poplar
Street and Fern is equivalent to the
angle between Elm Street and Fern
street. If the distance from E to F is
8 miles, F to P is 10 miles, and
other distances are indicated
below, how far apart is are points
M and P?
Name: ____________________________________
Geometry Per. _____
8-5 Homework
Date: ___________
1. Use the angle bisector theorem to find the missing side length of the segment in the triangle below:
CA = 12, CD = 6, BA = 15, DB = ?
2. What is the length of Wx in the triangle below, given that WZ = 24, ZY = 12, XY = 15.
3. In the diagram below of
, D is the midpoint of
, O is the midpoint of
If AC = 20, AT = 36, and CT = 22, what is the perimeter of parallelogram CDOG?
1) 42
2) 50
3 ) 78
)
4) 32
, and G is the midpoint of
.
4. Factor: 8y3z + 16xy
5. Solve: z2 – 2z = 24
6. Is AD an angle bisector? Explain why or why not.
Follow up: can we identify a shortcut method that would allow us to conclude that ∆𝐴𝐵𝐷 ~∆𝐴𝐶𝐷?
7. Using properties of similarity and specific vocabulary we’ve learned in this unit, describe why the two triangles
below are similar to each other.