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

Prisms is a solid with bases that are 2 congruent polygons. The other sides of the prisms are called the lateral faces.

Surface Area : Lateral area + area of the two ends(bases)
Lateral area: Perimeter of the base*height
Surface area : perimeter of base* height + 2(Area of the base)

Cylinders


Surface Area: Areas of top and bottom +Area of the side
Lateral area: 2*pi*r*h
Surface area: Surface Area = 2(pi r ²) + (2 pi r)* h

http://www.mathsteacher.com.au/year7/ch09_polygons/06_polyhedra/Image11220.gif
http://cf.ydcdn.net/1.0.0.30/images/main/A4cylind.jpg

WHAT IS THE SURFACE AREA OF A SPHERE?

how do we do compositions of transformations?

• Compositions of transformations are when 2 or more transformations are combined to form a new transformation.
• In a composition, you do the second one FIRST.

          SOMETIMES Written and ALWAYS rule:



EXAMPLE:

 

 

Sources:

http://www.regentsprep.org/Regents/math/geometry/GT6/composition.htm

how do we do the compositions of 3 transformations?

How do we define circles?

What is a circle?

• the set of all points in a plane at a given distance from a given point
• Given distance is called a RADIUS
• Given point is called a CENTER

Chord

• A line segment that connects two points on a circle



Diameter

• A line segment through the center of a circle with endpoints on the circle.
• Longest chord in the circle

Tangent

• A line that touches a circle in exactly one point.
• Two circles can be tangent at the same line and same point.

Arc

• An arc is a portion of the circumference of a circle.

Resources:

• http://www.mathatube.com/images/circle-lines_1_.gif

Any other lines or segments in a circle?

How do we calculate distance?

To calculate distance from two endpoints, you use the formula:

Distance Formula:


Example: Find distance between the two points : (-1,-2) and (9,-10)
1st step :use the distance formula

2nd step : square root* of (-1-9)^2+ (-2-(-10))^2
3rd step: square root of (-10)^2+(8)^2
4th step: square root of 100+64
5th step: square root of 164
6th step: 2 square root of 41
*square root:

Resources:
http://0.tqn.com/d/math/1/0/A/1/distance.gif
http://www.moomoomath.com/distance.jpg

Can you use this distance formula to calculate distances in shapes?

What are the special parallelograms?

A rhombus is a parallelogram with four congruent sides and angles

- The diagonals of a rhombus are perpendeicular
- The diagonals bisect the angles
- They bisect each other


A rectangle is a parallelogram with four congruent angles also called an equiangular parallelogram

-The diagonals of a rectangle are congruent ands bisect each other

A square is an equilateral rectangle or regular quadrilateral or an equiangular rhombus

-A square's diagonals are congruent, perpendicular & they bisect each other.


Resources:
http://www.analyzemath.com/Geometry/rhombus_1.gif
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg33UNJHn2qgmKLlEmfdl5gWiTqg0So6hdwpnVIRRolCI0UH9mwuPEoAus_ZS4N8xf0WvRyEGnlT32YVvap1K_IBXX252UFw9gHPG3LWsDWa9QtAn0bu6XSTzaweBK8oPRZ6_Qqi8rFma1y/s1600/rectangle+18.PNG
http://www.google.com/imgres?um=1&hl=en&sa=X&tbo=d&biw=1366&bih=618&tbm=isch&tbnid=ZdYwe4YDYLWyGM:&imgrefurl=http://www.geom.uiuc.edu/~dwiggins/conj28.html&docid=L8um8FYz5pA4mM&imgurl=http://www.geom.uiuc.edu/~dwiggins/proof29.GIF&w=364&h=196&ei=LWj5UOXEL6mB0QGooIDIBw&zoom=1&iact=hc&vpx=2&vpy=179&dur=1887&hovh=156&hovw=291&tx=225&ty=48&sig=114690899695513863037&page=1&tbnh=137&tbnw=256&start=0&ndsp=18&ved=1t:429,r:0,s:0,i:122

Are there any other special shapes?

How do we use the properties of parallelograms?

Parallelogram: A quadrilateral with 2 pairs of parallel sides.

 Properties

• The opposite sides are congruent.  
• Consecutive angles are supplementary. Example: 
• The opposite angles are congruent. 
• Diagonals bisect each other 
• Not always congruent
• Not always perpendicular 

http://www.wyzant.com/Help/Math/Geometry/Quadrilaterals/Properties_of_Parallelograms.aspx
http://ef004.k12.sd.us/ch8notes.htm
http://www.ck12.org/user%3AYWtlZWxlckBhY2VsZnJlc25vLm9yZw../book/ACEL-Geometry-2012-2013/r2/section/6.5/Proving-Quadrilaterals-are-Parallelograms/

How do we use properties of trapezoids?

12.7.12

First thing to know is what is a trapezoid.

• A trapezoid has exactly one pair of parallel lines

 

Properties of trapezoids

1. Trapezoid Consecutive Angles Conjecture                                                                                         -The consecutive angles between the bases of a trapezoid are 180 degrees (supplementary).

     2. Isosceles Trapezoid Conjecture 

         - The base angles of an isosceles trapezoid are congruent.

<A is congruent to <D.

Try solving this:  Find x.

  

1st step: 2x+5=3x+2

2nd step:subtract 2x from both sides

                                                    Equation should look like this: 5=x+2

3rd step: subtract 2 from both sides and you get your answer.

                                                    Answer is 3+x

* You don't do this to 4x-2 and 5x+3 because those are parallel not congruent.

1st picture from http://www.google.com/imgresum=1&hl=en&tbo=d&biw=1517&bih=693&tbm=isch&tbnid=sGz7fgFw41GoMM:&imgrefurl=http://www.ck12.org/concept/Trapezoids---Intermediate/&docid=eeMQ8dVhvkcDNM&imgurl=http://www.ck12.org/flx/show/image/user%25253Ack12editor/201209241348511282764513_dff7a8f97ef56047035788567c45fc33-201209241348512460267699.png&w=708&h=511&ei=MabCUMukGPC10AGlqYG4Aw&zoom=1&iact=hc&vpx=4&vpy=274&dur=292&hovh=191&hovw=264&tx=128&ty=97&sig=108818101793549926599&page=2&tbnh=145&tbnw=202&start=27&ndsp=38&ved=1t:429,r:42,s:0,i:215

2nd picture from http://www.geom.uiuc.edu/~dwiggins/conj22.html

  

WHAT ARE THE RHOMBUS'S CONJECTURES?