Polygons
What are the two main criteria that make a polygon a polygon?
1. Straight Sides
2. Completely closed
Evaluate: 3x+8; x=5
3(5)+8
15+8
23
Triangle Angle Sum Theorem
The three internal angles of a triangle sum to 180 degrees
This person famously stated, just prior to his death, "Leave me to my circles." Gave rigorous proofs of the area of a circle
Archimedes
For each new side of a polygon, how many internal degrees of measurement are added to the polygon?
180 degrees
Evaluate: x^2+3x-1; x=8
(8)^2+3(8)-1
64+24-1
64+23
87
Triangle Inequality Theorem
Two sides of a triangle angle added together are greater than or equal to the length of a third side.
This person developed a method for solving the quadratic through completing the square; A mathematical discipline bears his process
Al-Khwarizmi
Name the polygon: two parallel sides; two sides that are congruent; quadrilateral
Isosceles trapezoid
Factor: x^2+10x+16
(x+8)(x+2)
Alternate Interior Angles
Two angles interior to the parallel lines of a transversal that are on alternate sides of the transversal are congruent
This figure is responsible for establishing a rigorous foundation for geometry, and by extension, a rigorous foundation for future mathematics.
Euclid
Name the polygon: Concave; five acute angles; five obtuse angles; composite of triangles and a pentagon
Concave decagon (star)
Find an equation for the line that passes through the points (3,5) and (5,8)
m=(8-5)/(5-3)
m=3/2
y-5=3/2(x-3)
y-5=3/2x-9/2
y=3/2x+1/2
Longest Side, Greatest Angle Conjecture
The longest side of a triangle is opposite the largest angle
This figure developed the now widely used x-y axis system for plotting the relationship between variables
Rene Descartes
Name the unique properties of squares
1. Parallel sides
2. Equilateral sides
3. Right angles
Find x: f(x)=2x^2+1; f(x)=11
2x^2+1=11
2x^2=10
x^2=5
x=+/-sqrt(5)
Pythagorean Theorem
a^2+b^2=c^2
The sum of the squares of the two shortest legs of a right triangle are equal the area of the square of the hypotenuse
This figure demonstrated that there are various sizes of infinity; responsible for the development of set theory
Georg Cantor