Similarity Review

What is Similarity?

Similarity is when two figures have the same shape but different size making them similar. An example of similar shapes would be the images below.



All three triangles are similar because they have the same angle measurements and same shape but they are different in size.



How do we solve similarity problems?

To solve similarity problems you could follow this simple tutorial:


http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&sqi=2&ved=0CH4QtwIwBg&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dtm-_6sFdfk8&ei=e4zWT7HYEM746QGi0uGlAw&usg=AFQjCNEGGFNJiYfqx9TKmH2MT_Ku13MgwQ&sig2=5-kVGK027FmUJCbNXJy6Cw

 All you have to do is create a proportion between the original figure and the dilated figure for example, try to find x in the image shown below:

How do we use right triangle similarity?

Right triangle similarity is when an altitude is drawn in a right triangle creating 3 separate triangles.
You can see how this is done in this slideshow:
http://www.slideshare.net/teacherfidel/right-triangle-similarity


To solve right triangle similarity problems is the same as other similarity problems where you first start by creating a proportion.

When are triangles similar?

Two triangles are similar when they have at least two angles of the same measure such as these:

Aim: How do we find the volume of pyramids and cones?

To find the volume of a cone you must use the formula (1/3) times pi times the radius squared times the height.





The volume of this figure would be 6 squared (36) time 15 (540) divided by 3 (180). You could leave this answer in terms of pi which would equal 180pi cm cubed.


To find the volume of a pyramid you have to use the formula (1/3) time base times height.

The volume of this pyramid would be 8 times 8 (36) times 9 (324) divided by 3 (108). The final answer would be 108 cm cubed.









Find the volume of theses two solids:

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

To find the volume of a sphere you must use the formula 4 times pi times the radius cubed divided by 3.

For example:

 

The volume of this sphere would be 4 cubed (64) times 4 (254). After this you could just leave the problem in terms of pi which is 256pi and  then divide it by 3. The final answer would be 85.3pi cm cubed.




To find the surface area of a sphere you must use the formula 4pi times radius squared.




 

Using the same sphere the surface area would be 4 squared (16) times 4 (64). After you find this you could just leave the final answer in terms of pi which is 64pi.




Find the surface area and volume of this sphere:

Aim: How do we find surface area and lateral aea of prisms and cylinders?

The formula to find the surface are of a prism is base times height times width

For example in this image the base length is 5, the height is 3  and the length is 10.
5 x 3 x 10 = 150

And to find the lateral are of this figure would would simply subtract the area of the two faces which would be the area of the two squares.
3 x 5 x 2 = 30
Then you would subtract the area of the two faces from the total surface area.
150-30= 120.
Lateral area = 120.

 Find the Surface and lateral area of this prism:


The surface area formula of a cylinder is: 2 π r² + 2 π r h



To find the surface area of his cylinder you would do radius squared (9) times 2 times π.
This will find the area of the two circles. To find the area of the rest of the solid you would have to do
6 x 3 x 2 x π. 
This formula would also solve the lateral area.

But to find the surface area you would have to add the outcome of the first equation and the second equation.


Find the surface and lateral area of this cylinder:


 

Aim: How do we idnetify solids?

Solid- 3- dimensional figures with width, depth and height.

All solids have volume and surface area.
There are two types of solids:
Polyhedra- having all flat surfaces.
Non- Polyhedra- if any surface is not flat.

All solids fall under these two categories.

Figures that would be considered polyhedras are; cubes, pyramids and prisms
Figures that fall under the category of non-polyhedras are figures like; cylinders or spheres.

Aim: How do we find the area of regular polygons?

Area of a regular polygon- The are of a regular polygon is n (number of sides) times a (apothem) times s (side length) divided by 2 (nas 1/2).







Fo example if a (apothem) is 6 cm long and the side (s) is 5 cm long the area of this hexagon would be
5 times 6 times 6, which equals 180 and then you divide 180 by 2. The are of this hexagon would then be 90 cm squared.




What would be the area of a pentagon with an apothem of 7 and a side length of 8?

Aim: How do we find the area of a circle?

Circle Area Formula- The area of a circle is given by the formula: pi times the radius squared.

So for example:




The area of this circle would be the radius (5) squared (25) times pi. You could solve this problem by typing it into a calculator or just leaving it in terms of pi, which would be 25 pi.



Solve the area of this circle:

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

To find the area of a parallelogram is the same formula as finding the area of a rectangle:
Base times Height (b x h) = Area

• 

To find the area of a kite you must follow this formula:







(Diagonal 1 + Diagonal 2) / 2 = Area




To find the area of a trapezoid you must follow this formula:









(Base 1 + Base 2) /2  times Height = Area





Try these area problems:





If Diagonal 1 is 25 and Diagonal 2 is 30, what is the area of the kite?




If Base 1 is 6, Base 2 is 3 and the height is 8, What is the area of the trapezoid?




If the base is 7 and the height is 5 what is the area of the parallelogram?

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

To calculate the area of a rectangle you must follow this formula:
 
Base times Height (b x h) = Area



To find the Area of a triangle you must follow this formula:

Base times Height times 1/2 (b x h x 1/2) = Area



Try these problems:
If the base of a rectangle is 7.3 and the height is 4, what is  the area?

If the base of a triangle is 6 and the height is 5, what is the area?

Aim: How do we find the locus of points?

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

To find the locus of points on a graph you would need to know the condition, for example;

What would the locus of points be 15 meters from Point P? Point P, from the theorem, is the stake to which Fido, the dog, is tied.  His leash is 15 feet long.  The path that Fido can travel at the end of his leash is "the locus of points". The locus of points at a distance of 15 feet from point P is a circle (with center P and radius 15).

In this image, Fido could move freely within the 15 meters of his leash, so any point at the end of his leash would satisfy the condition asked.

There are other conditions such as lines that can include a locus of points.
The locus of points for a single line would be two separate lines on the sides.
 For example what are the locus of points 3 cm from line L, would be the two red lines.

Try this: what are the locus o points from X=2?

Aim: What is a mathematical statement?

A mathematical statement is a statement that can be judged to be true or false.

Conditional- The conditional is the most frequently used statement in the construction of an argument or in the study of mathematics.

Converse- Reversing the first and second statements in a conditional.

An example of a Conditional: If you take a shower, than you smell good.
The converse of this statement would be: If you smell good than you took a shower.

Than there is the Inverse of a conditional.

Inverse- Formed by negating the hypothesis and conclusion.
The inverse of the conditional would be: If you don't smell good than you didn't take a shower.

Write the converse and inverse of this statement: If it is sweet, than it has sugar.

Aim: How do we solve logic problems using conditionals?

Conditional- asserting that the existence or occurrence of one thing or event depends on the existence or occurrence of another thing or event; hypothetical.
an example of a logical conditional in math would be: If there is a fire, than there would be smoke.


If you want to solve problems such as what is the contrapositive of this statement you would have to change certain parts of the sentence.
Contrapositive- the inference drawn from a statement and negating the terms and changing the order.


The contrapositive of the conditional example would be: If there is no smoke, than there is no fire.


Find the contrapositive of this statement: If the television is on than electricity is being used.

How do we review transformations?

Most or almost all transformations fall under four categories: Reflection, dilation, translation, rotation.

To reflect an image you need a line to reflect over. You can reflect any image over any line.
For example: reflecting points (3,2) (5,1) and (4,5) over the y-axis would be (-3,2), (-5,1) and (-4,5).

To dilate an image or point you would need a scale factor. the scale factor is what you multiply the point's coordinates by. For example if the scale factor is 2 and the points are (5,4) the new point coordinate would be (10,8).


Translation, which is probably the easiest out of all the transformations, is when you move the point accordingly. For example if it says T (2,3) and the point is (4,6) then the new translated point is (6,9).


A Rotation is a transformation that turns a figure about a fixed point. If you nrotate any point it will probably be counter-clockwise unless requested to be rotated clockwise. If you have a point (1,6) and rotate it 90 degrees then it would be (-6,1). when rotating you could always follow this rule.
90 degrees: (A,B)--->(-B,A)
180 degrees: (A,B)--->(-A,-B)
270 degrees: (A,B)--->(B,-A)

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

Other definitions of transformations would be things such as; Glide reflections, orientation, isometry and invarient.

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

Orientation- orientation refers to the arrangement of points, relative to one another, after a transformation has occured.

Isometry- an isometry is a transformation of the plane that preserves length.

Invarient- a figure or property that remains unchanged under a transformation of the plane is reffered to as invarient. No variations have occured.

All of these terms are other kinds of defintions of transformation.

One of the more important and more used terms here would be isometry. Every kind of transfromation falls under the definition of isometry except for dilation.

Aim: How do we identify composition of transformations?

Composition of Transformations- When two or more transfromations are combined to form a new transformation.

An example would be this:

Whichever transformation has the little circle in front of it is the one that is done first
because it is read as the reflection  over the x-axis after the translation of (3,4).


Try this: R y-axis after T (7,8) on point (2,3).

Aim: How do we graph dilations?

Dilation's- A dilation is a type of image that causes an image to stretch or in proportion to its original size.

When a image is dilated it has a scale factor in which it either shrinks or enlarges.if the scale factor is >1 then the image is enlarged. If the scale factor is <1 then the image will shrink. Whatever the scale factor is then you multiply it by each of the points.


 This image is an example of a dilation. 
The second triangle increased proportionally 
from the original shape.




Dilate these points if the scale factor is 2:
(4,3), (-1,2), (0,5)

Aim: How do we graph transformations that are reflections?

A reflection is a transformation that creates mirror images over the X or Y-axis:
Ex. This image is an expample of a refelction.



To reflect certain points on a graph you have to follow this rule:
Reflecting over the X-axis-- (x,y) ---> (x,-y)
Reflecting over the Y-axis-- (x,y) ---> (-x,y)


Try this: Reflect point (6,9) over the X-axis.