How Do We Find Surface Area and Lateral Area of Prisms?

A Prism is a 3-D shape that has two parallel faces, also known as bases, that are as well congruent polygons. Each of the faces of the prism is a rectangle (Right Prism) and in geometry they are known as Lateral Faces. The joining edges and faces are perpendicular to the base faces. 


Finding the Surface Area of a prism is very simple. All you do is add the area of the lateral faces with the areas of the bases. 


S.A.: 2LW+2LH+2HW
L.A.: ph


L: length
H: height
W: width
p: perimeter of the base
h: height 

How Do We Identify Solids?

In geometry there are a whole bunch of different shapes. There is a term called SOLID GEOMETRY. Solid Geometry is the geometry of 3-D shapes. There are two types of solids: Polyhedra and Non-Polyhedra 


 -Polyhedra: a 3-D solid figure in which each side has a flat surface. Those surfaces are polygons that are joined at their edges. A Polyhedron has NO CURVED SURFACES!!!! 
          • Called regular if the faces are Congruent 
          • Also if the Regular Polygons and the same number of faces meet at each vertex.
 Example: Pyramid, Prism. 


 -Non Polyhedra: a 3-D figure that has cures in their shape. 
 Example: Cylinder, Cone, Sphere.

How Do We Find the Area of Parallelograms, Kites, and Trapezoids?

AREA FOR:

• Parallelogram: Base * Height
• Trapezoid: (Base 1 + Base 2) * Height                                 

                                            2

• Kites: Diagnal 1 * Diagnal 2 / 2

What is Locus?

Locus Point: the setof all points that satisfy a given condition. A general graph of a given equation.

Two Fixed points; a line through the middle of the points. Perpendicular Bisector
One Line; two parallel lines on opposite sides of the original line
One Point; forms a circle
Two Intersecting Lines; two intersecting lines halfway between the two original lines

• 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 the 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 i another line, halfway between both lines, and parallel to each of them.

Loucs Rule:

1. 1 Point- Circle
2. 2 Points- 1 Line
3. 1 Line- 2 Lines
4. 2 Lines- 1 Line
5. 2 Intersecting- 2 other intersecting

How Do We Find Compound Loci?

Compound Locus: Problome involves two or more locus conditions occurring at the same time. the different conditions in a compound locus problem are generally seperated by the word "AND" or the words "AND ALSO".

What Is Logic?

LOGIC IS THINKING!!!!!!

Everyday, we use logic. Logic is used to determine if something is true or false.

Four Conditionals:

1. Conditional
2. Inverse
3. Converse
4. Contrapositive

How Do We Use the Other Definitions of Transformations?

Besides the common types of transformations, there are five more types of transformations.

• Glide Reflection: the cmbination of a reflection in a line and a translation along the line.
• Orientation: refers to the arrangement of points, relative to one another after a transformation has occurred.
• Isometry: a transformation of the plane that preserves length. (an opposite isometry change the order such as clockwise changes to counterclockwise).
• Invariant: a figure or property that remains unchanged under a transformation of the plane is refered to as invarients. No variations have occurred.
• Direct Isometry (with orientation is the same): has both isometry amd same orientation (rotations).

How Do We Identify Composition of Transformations?

Sometimes, a math question may ask that you create a composition of transformations. But what exactly is that? A Composition of Transformations is when two or more transformations are combined to form a new transfromation.

Example:

• Rotate the figure 90 degrees clockwise and then reflect the rotated figure over the x-axis
• Translate the figure 5 units up and 4 untits to the right. Now dialate the new figure with a scale factor of 2.

Those two examples are examples of a composition of transformations.

How Do We Graph Dilations?

Out of all the types of transformations, dilation is probably the easiest one. To start off, Dilation is a type of transformation that causes the image to either enlarge or shrink from its original size. The dilation is all based on a Scale Factor. A Scale Factor is the ratio by which the image enlarges or shrinks.

• If the scale factor is greater than 1 (>1), then the image is enlarged.
• If the scale factor is greater than 0and less than 1 (>0 and <1), then the image shrinks.

To actually find the exact coordinates of the new image, you multiply each point (both x and y), by the scale factor and once you find the answer, you plot the new points on the graph.



This is an example of a dilated figure. The original figure is the pink triangle and the new figure is the blue one.



How Do We Graph Rotations?

Unlike rotation and reflection, where you are just sliding the figure around the graph, Rotation involves rotating not only the figure, but the graph as well. In Rotation, there are three different ways you can describe a rotation:

1. An angle of rotation.
2. A direction: Clockwise or Counterclockwise (If for example the question does not specify which way to rotate the figure, you always rotate it Counterclockwise).
3. A center of rotation.

There are three different ways to rotate a figure on the graph:

1. 90 Degrees; (A,B)----> (-B,A)
2. 180 Degrees; (A,B)---> (-A,-B)
3. 270 Degrees; (A,B)---->(B,-A)

How do we graph transformations that are reflections?

The Transformation Reflection is when a figure is reflected over the y-axis or  the x-axis. In Geometry there are two types of reflections, the first one is Reflectional Symmetry and the other one is Line Symmetry. 

Reflectional Symmetry: this type of symmetry is when the figure can be evenly divided through the middle. This means that it is the same exact size on both side. 

Line Symmetry: this symmetry is when the figure is transformed to the other side of the axis and creates a mirror image. 

In Reflection there are two rules. One is for the x-axis and the other is for the y-axis...


X-Axis: (x,y)------>(x,-y)
Y-Axis: (x,y)------>(-x,y)

How do we identify transformations?

A transformation is when a geometric shape is moved on a graph. These transformations can be identified four different ways. These ways include: Translaton, Reflection, Rotaion, and Dialation.

Translation: When a figure is translated, the geometric shape is simply moved the same distance from the axis line to the opposite side.

Reflection: The figure is relected over the line of symmetry or you simply count how many spaces the figure is away from the axis line.

Rotation: The figure is only turned around at a single point. its moved by degrees. 

Dialation: Last but not least, Dialation is when the size of the figure is either enlarged or reduced.