GeoBlog / 2006-07 / 2nd Period

Polyhedron
3-D solid formed only by polygons
faces- polygonal sides of a solid
edges- segmental boundaries of each face
vertices- points where edges meet

*What solids aren’t polyhedrons?
usually formed with circles and/or smooth faces
Examples: cones, spheres, and cylinders

The Types of Polyhedrons

*Prisms

    -contain 2 bases (named for the bases)

    -side faces are rectangles

*Pyramids

    -contain one base (named for the base)

    -side faces are triangles that meet at a vertex opposite of base

*Platonic Solids

    -all faces are made of regular polygons

    -named for the number of faces

    -There exist only five made of only triangles, squares, and pentagons

    -known as regular polyhedrons

(The five platonic solids)

*Tetrahedron

    -4 equilateral triangles

    -tetra- 4-sided

    -hedron- 3-D solid

    -aka triangular pyramid

*Hexahedron

    -6 squares

    -hexa- 6- sided

    -hedron- 3-D solid

    -aka cube or square prism

*Octahedron

    -8 equilateral triangles

    -octa- 8-sided

    -hedron- 3-D solid

    -same as 2 square pyramids placed together

*Dodecahedron

    -12 regular pentagons

    -dodeca- 12-sided

    -hedron- 3-D solid

*Icosahedron

    -20 equilateral triangles

    -icosa- 20-sided

    -hedron- 3-D solid

A trapezoid is a quadrilateral with exactly ONE pair of parallel sides.

**In a trapezoid, the midsegment is ½ the sum of the bases and is ½ the sum of the bases. 

**In an isosceles trapezoid, diagonals are congruent.

**In an isosceles trapezoid, base angles are congruent. 

EXAMPLE 1:
QRST is a quadrilateral with vertices:
Q(-3,-2) R(-2,2) S(1,4) T(6,4)
Verify that QRST is a trapezoid. 
Do this by finding lines that are parallel.
(use formula for slope)
QR= -2-2/ -3–2
QR= 4  RS= 4-2/1–2
RS=2/3  
QT= 4- -2/6- -3
QT=2/3

ST= 4-4/6-1
ST=0 So, this is a trapezoid because RS and QT are parallel. 

Now use the distance formula to find out if this is an isosceles trapezoid. 

QR=√ (-2-2)+(-3- -2)
QR=√17 
 

ST=√(4-4)+(6-1)
ST=5 

Tonights homework is page 442 problems 9-19. Good Luck!

If you look at the relationship between the faces, edges, and vertices of a polyhedron, you see that the sum of the faces and edges is equal to 2 more than the number of edges. Euler turned this fact into a formula.

Euler’s Law:
F+V=E+2
(F= faces E= edges V= vertices)

Examples:
1) A prism has 12 faces and 6 vertices. How many edges does it have?
F+V=E+2
12+6=E+2
18=E+2
E=16

2) A pyramid has 5 faces and 9 edges. How many vertices does it have?
F+V=E+2
5+V=9+2
5+V=11
V=6

3) A polyhedron has 13 edges and 7 vertices. How many faces does it have?
F+V=E+2
F+7=13+2
F+7=15
F=8

Polygon / Polyhedron TEST Tomorrow! 

Chapter 1 – Section 6
Chapter 8 – Section 1
Chapter 12 – Section 1

Test Specs:
Definitions     (30 pts)
Identify Polygons     (15 pts)
Perimeter    (10 pts)
Interior / Exterior Angles     (15 pts)
Identify Polyhedrons     (16 pts)
Apply Euler’s Law     (14 pts)

HINT:  Identify Polygon & Polyhedron will be fill-in-the blank.  Know these shapes and their attributes by both description and drawing.
I’ll check in at 7:30 and 10:00 for questions. 
Mr. Carrigan be in the classroom at 7:00 tomorrow morning. (I have a parent conference.)

Study!
Mrs. C

Quadrilateral TEST Tomorrow! 

Chapter 8

Test Specs:
Characteristics (like quiz)     (26 pts)
Algebraic Applications   (44 pts)
Coordinate Applications    (30 pts)

HINT:  One of the coordinate questions will include writing your explanation of how and why you got your answer. 

Answers to review work is posted on the Homework Page.
http://www.shelbyed.k12.al.us/schools/omhs/faculty/sculbreth/hmwrk.htm

I’ll check in at 10:00 for questions. 
Mr. Carrigan be in the classroom at 7:00 tomorrow morning.

Study!
Mrs. C

Parallelogram Characteristics:

-opposite sides parallel

-opposite sides congruent

-opposite angles congruent

-adjacent angles are supplementary

-the diagonals bisect eachother

Rectangle Characteristics:

-diagonals are congruent

-four right angles

Rhombus Characteristics:

-diagonals bisect the angles

-all sides are congruent

-diagonals are perpendicular

Square Characteristics:

-all characteristics of both rhombus and rectangle

 Formulas:

a²= b²+c²-2bc(cos<A)

b²= a²+c²-2ac(cos <B)

c²= a²+ b²-2ab(cos<C)

Use the law of Cosines when:

-you’re given three sides

-you’re given two sides and and <

Ex 1:

x=? y=11 z=25 <x= 45

x²=25²+11²-2(25)(11) cos 45

x²= 625-2(275) cos 45

x²= 625-550 (cos 45)

x= 18.9

Ex 2:

<L=?

24²=5²+27²-2(5)(27) cosL

576=25+729-770 (cosL)

178=270(cosL)

<L= 48.8

If parallelogram:
**(both) If opposite sides are parallel, then p’gram
**(both) If opposite sides are congruent, then p’gram
**(both) If opposite angles are congruent, then p’gram
**If diagonals bisect, then p’gram
** If one pair of opposite sides are parallel and congruent, then p’gram

Example 1:

Determine whether this quadrilateral is a parallelogram. Justify your answer.
Yes, if opposite angles are congruent, then p’gram.

Example 2:
 Yes, if diagonals bisect, then p’gram.

-Tonights homework is Pg. 421 #13-24 and Pg. 769;8-3 #1-6
 

Parallelogram: A quadrilateral where opposite sides are parallel

Characteristics

1. Opposite sides are parallel
2. Opposite sides are congruent
3. Opposite angles are congruent
4. Consecutive/ Adjacent angles are supplementary
5. Diagonals bisect each other

<>** In a parallelogram, the diagonal divides the parallelogram into two congruent triangles

The triangles are congruent by ASA or SSS

<>

<>Example one: RSTU is a parallelogram. Find angle URT, RST and y.
angle URT equals 40 degrees because of alternate interior angles
angle RST equals 122 degrees because consecutive angles are supplementary
y equals 6 because 3y=18, set them equal because opposite sides are congruent

<>Example two: Find angle BDC , angle BCD, and x
x= 5 because you set 4x equal to 20 because opposite sides are congruent
angle BDC equals 54 degrees because alternate interior angles are congruent
angle BCD equals 62 degrees because consecutive/ adjacent angles are supplementary

The homework is page 415, problems 16-31 and 35-36

An angle of elevation is the angle between the line of sight and the horizantal when a observer looks upward.

An angle of depression is the angle between the line of sight when an observer looks downward, and the horizontal.