Aim: How do we use scant, chord and tangent ratios in cirles?

-If Two cords intersect inside a circle, then the prodect of the segements of one cord equals the product of the segements of the other chord.

-Theorem 84: If two secant segements intersect outside a circle, then the product of the secant segements with its external portion.
WO=WO

How do you find the surface area and volume of a sphere?

Aim: How do you find the surface area and volume of a sphere?

SPHERES
- A sphere is the set of all points in the space equidistant from given point called the center.


Formulas:
SA= 4(3.14r^2)
LA= 4(3.14r^2)

For Example:
1. find the Surface area of the Sphere
r=17
formula=4(3.14r^2)

Step 1: write the formula if you want
EX: formula=4(3.14r^2)Step 2: Plug in the given information into the formula
EX: SA=4(3.14r^2) R=17
SA:4(3.14*17^2)
Step 3: Solve the problems
EX: 4(3.14x17^2)
        4(3.14x289)
        4(907.9)
      SA=3631.6
Try on your own:
Solve for SA


R=13

How do we find the surface area and lateral area of pyramids and cones?

Aim: How do we find surface area and lateral area of pyramids and cones?
Do Now: Reviewing Triangles
1. Find the area of the triangle
height is 20 and the base is 16



Pyramid
A pyramid is a solid that connect a polygon base to point. The Pyramid has 3 triangle faces and a base that is a square.

The Slant Height of a pyramid is the height of one of the lateral faces.

* Formulas*
Lateral area= (1/2)pl
SA= L.A+B

Solve for lateral area and surface area

How do we find the area of parallelograms, kites and trapezoids?

Aim: How do we find the area of parallelograms, kites and trapezoids?

Area of a Trapezoid is the 1st base + the 2nd base and then you divide by 2 and then times the height given in the trapezoid.
Formula: b1+b2/2*H 

Try it out:
1.  Base 1=14
Base 2=12
The Height=6






Area of a Kite is the first diagonal times the 2nd diagonol divided by 2.
Formula: d1*d2/2

Try it:
1. Diagonal 1=7
Diagonal 2= 16

How do we calculate the area of rectangles and triangles ?

Aim: How do we calculate the area of rectangles and triangles ?

Triangle Area Conjecture= (1/2)bh
The Area of a triangle is given by the formula A=(1/2)bh. A is the Area, B is the length of the base, and H is the height of the triangle.

Example:

 The Triangle ABC.
The Length of AC is 9mm and the length of AB is 20mm.

Step1: Know the given information, The problem gave you already that the length of the base is 9mm and also that the height is 20mm.

Step2: plug the new information in the formula A=(1/2)bh 
Ex: A=(1/2) 9 X 20

Step 3: Multiply The base and the height
Ex: 9 X 20= 180

Step4: Once you got your answer divide the answer by 2
180/2= 90

And the answer to the problem above is 90mm^2

The Elegant Locus

 Aim: The Elegant Locus


Definition: The locus is the set of all points that satisfy a given condition.


- A locus is a general graph of given equation.


- The locus of points equidistant from a single point is set of point, equidistant from the point, in every direction.


- The locus of points equidistant from a single point is a set of points, equidistant from the point, in every direction.

- The Locus of points equidistant from two points is perpendicular bisector of the line segment connecting the two points.

- The locus of points equidistant from a line are two lines, on opposite sides, equidistant and parallel to that line.

- The locus of points equidistant from two parallel lines is another line, half way between both lines, and parallel to each of them.

-The locus of points equidistant from two intersecting lines are two additional lines that bisect the angles fromed by the original lines.

Try now: From Regentsprep.org

The locus of points equidistant from two intersecting lines is ...

Choose: one circle two parallel lines two circles a pair of angle bisectors

Aim: How do we identify solids ?

Aim: How do we identify solids ?

    Solid Geometry: Solid geometry is the geometry of 3-dimensional space.

Properties:

• Volume
• Surface Area

Two Main types of solid:

1. Polyhedra
2. Non-Polyhedra

Polyhedra: Flat Surface

• Prism                       
• Pyramids
• Platonic solids

Non-Polyhedra: If any surface is not flat

• Sphere
• Torus
• Cylinder
• Cone

Prisms: Has the same cross section all along its length
Cross sections: The shape you get when you cut straight down
Surface Area: The area of the bases + the Lateral Area ( area of the side)

What is a mathematical statement?

Aim: What is a mathematical statement?

Definitions:
A Mathematical Statement: Is a statement that can be judged to be true or false
 For Example:

1. All quadrilaterals have 4 sides. True or False
2. 17-3=14. True or False

Negations: The word "NOT" Always have the opposite truth value.
 Such as: " If its not raining, Then I will not take my umbrella.
 The opposite: If it is raining, Then I will take my umbrella.

When a Statement has "AND" or "OR"
 Both Statements are true, and also can be either statement can be true.

 Conditional: The conditional is the most used statement in the construction of and argument or in the study of mathematics.
For Example:
Conditional: If its raining, Then its cloudy.
Argument: If it is not cloudy, Then it is not raining,

Inverse: Is formed by negativing the hypothesis and conclusion.
EX:
"If Talia had 14 balloons after giving 7 away, Then Talia has 7 left."
Inverse: "If Talia has 7 balloons left, Then she originally had 14 balloons before giving 7 balloons away."

Solve:

"If Sally's age is twice as Billy's age and Billy's age is 3, Then Sally's age is 6."
What is the Inverse?

Inverse:

How do we graph rotation

Aim: How do we graph rotation?

Rotation

1. An angle of a rotation
2. Direction is counterclock wise or clockwise
3. center of rotation

90 degrees
(A,B) ---> (-B,A)

180 degrees
(A,B)---->(-A,-B)

270 degrees
(A,B)------>(B,-A)

Rotated 180 degrees

Rotated 270 degrees

How do we use the other definitions of transformations

Aim: How do we use the other definitions of transformations ?

Definitions

Glide Reflection: the combination of a reflection in a line and translation along that line.

Orientation:  An arrangement of points relative to one another, after a transformations has occurred.

Isometry: is the transformation of the plane that preserves length.

Invariant: A figure or property that remains unchanged under a transformation of the plane is referred to as ivanant. No variations have occurred.

How to identify composition of transformations

Aim: How do we identify compositions of transformations?

Composition of Transformation
-When 2 or more transformations are combined to from a new translation, in the result is called the Composition of Translation.

Example:

In this image you can see that the traingle was moved from its orignal spot to being reflected off the Y-axis first and then from there is reflected off the X-axis.

You could do any type of combination with these Translations

-Reflection

-Dilation

-Translation

-Rotation

How Do we Graph Dilations?

Aim: How do we Graph Dilations?

A Dilation is a type of transformation that casuse an image to stretch or shrink. It could decrease in size or increase.

•  The scale Factor is the ratio in which the image stretch or shrinks.
• If the factor is greater than 1, then the image will get bigger than the original image.
• If the factor is greater than 0 but less than 1 the image will shrink form the original.
• Multiply the dimensions of the priginal by the scale factors to get the new dimensions of the dilated image.

 Example:
A(-2,-2) --> A'(-4,-4)
B(-1,2) --> B'(-2,4)
C(2,1) --> C' (4,2)
D2

In this figure, it was dilated by 2 from its original image and each point is multiplied by 2.

Some times the image will shrink and this is when the image is dilated by a ratio like

ex: D(1/3) the points will be change from its original to something smaller.

Now try it out:



What are the coordinates of the image of point B under a dilation with center at the origin of scale factor 1/3?

Choose one:
(-1,-3)
(-1,-1)
(-9,-3)
(-9,-9)

How do we identify Transformations?

Aim: How do we identify Transformations?

Transformations are when you move a geometric figure.

There are 4 transformations
-Rotations
-Dilations
-Reflexions
-Translations

-A Translation is when every point is moved the same distance and same direction.
Example: From each point of the triangle move Five units to the right and 3 units down.


-A Dilation is the enlargement or reduction in the size and image.
Example:


-An Rotation is when a figure is turned around a single point.
Example: Move each point 90 degrees Counterclockwise.

- A Reflection is when a figure can be flipped over a line of symmetry.
Example:

Question:

	This graph illustrates a reflection over the x-axis. True False
	Explanation If you folded the graph along the x-axis, the figures would coincide.