GeoBlog / 2006-07 / 2nd Period

RELATIONSHIP                                             THEORM

1. 2a=b                                                       1. **The radius is  
   a=1/2b                                                       1/2 the diameter.   
     

    

2. YZ is congruent to WX                       2. **Congruent chords
   arc YZ is congruent to arc WX               form congruent arcs.
3. TU is perpendicular to RS                 3. **If a diameter is
   e is congruent to f                                   perpendicular to a
   arc SU is congruent to arc UR              chord, then it bisects
                                                                     the chord and its arc.
4. NO is congruent to QP                       4. **Congruent chords are
   g is congruent to h                                    equidistance from the
                                                                       center of the circle.                                            

Circle- The locus of all points in a plane equidistance from a given point.

Center- Given point/how you name the circle.

Circumference- distance around the circle.

Chord- segment w/ endpoints on the circle.

Diameter- Chord through the center.

Radius- segment w/ endpoints on the circle and center. equals 1/2 of the diameter.

Pi= Circumference/ diameter

Circumference= Pi(diameter)

**Interior angle formed by 2 chords=1/2 the sum of the intercepted angle

**Exterior angle formed by 2 secants=1/2 the difference of the intercepted arcs

**Exterior angle formed by a secant and a tangent=1/2 the difference of the intercepted angle

**Angle formed by a tangent and secant to the point of tangency=1/2 the intercepted arc

**Exterior angle formed by 2 tangents=1/2 the difference of the intercepted arcs

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

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