The tallest inhabited building in the world is the Sears Tower in Chicago. If the observation tower is 1450 feet above the ground level, how far can a person standing in the observation tower see? Use 3960 miles for the radius of Earth. Example:

a^2+b^2=c^2

4^2+3^2=c^2

16+9=c^2

25= c^2

5=c Pythagorean Theorem: in a right triangle, the square of the lengths of the legs.

a^2+b^2=c^2

Right Triangle Hypotenuse By: Hanleigh Review of Algebra & Geometry Sets: Real numbers or elements inclosed within a pair of braces.

example: {0,1,2,3,4}

Set-builder notation: D={ x | x is a digit}

Example: E={x | x is an even digit}={0,2,4,6,8}

Real Numbers are represented by symbols such as 25,0,1\2,-5/4

Rational Numbers: a number that can be expressed as a quotient a/b of two integers where the integer b can not be 0

Example:3/4,-2/3

terminating decimal

non-terminating with repeating blocks of digits

Irrational Numbers: Real numbers that are not rational

decimals that neither repeat nor terminate

Example: pi, 1.414213... Algebra Review Algebraic Expressions

Variable: a letter used to represent any number from a given set of numbers

Constant: either a fixed number , such as 5 or 1/2, or a letter that represents a fixed number

5x

Example: x+3y and 5xy, if x=3 and y=-1 Distributive Property

a(b+c)=ab+ac Zero Product Property If ab=0 then either a=0 or b=0 or both=o 2x=0

2=0

x=0 Geometry Review Leg Leg a) x+3y

3+3(-1)

3-3

0 b) 5xy

5(3)(-1)

-15 How to find the Hypotenuse Constant variable 4 3 Real Number Lines Applying the Pythagorean Theorem 0 1450 feet origin 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 P Coordinate Graphing Inegualities a b a<b b a b<a a

b a=b a<b, a>b, a=b, a>b, a<b inequality symbols d 1450 feet inequalities: a statement in which two expression are related by an inequality symbol strict inequalities: statement on the form a<b or b>a

~nonstrict inequalities: a<b or a>b 3960 miles a>0 is equivalent to a is positive

a<0 is equivalent to a is negative use parentheses to indicate the number is NOT part of the graph us a bracket or a closed circle to indicate that the number is part of the graph Distance Between Points If P and Q are two points on a real number line with coordinates a and b, respectively, the distance between P and Q, denoted by d(P,Q), is

d(P,Q) = I b-a I 0 1 2 3 4 5 -1 -2 -3 -4 -5 d(P,Q) = I (-5)-4 I = 9 d(P,Q) = I (-2)-4 I = 6 Exponents If a is a real number and n is a positive integer, then the symbol a represents the product of n factors of a n 5 3 exponent base = 5 x 5 x 5 If a = 0 and if n is a positive integer, then we define

a = a if a = 0 -n 1/n Law of Exponents a a = a a /a = a = 1/a (if a = 0) (a ) = a (ab) = a b (a/b) = a /b , if b = 0 m n m+n 5280 SOLUTION: m n m-n n-m From the center of the earth, draw 2 radii one through the sears tower and the other to the farthest point a person can see from the tower. Use the pythagorean theorem.

Since 1 mile=5280 feet, then 1450ft.=1450 miles d^2+(3960)^2=(3960+1450/5280)^2

d=(3960+1450/5280)^2-3960^2 2175.08

d 46.64 miles Area=L x W Perimeter=2L+2W Geometry Formulas m n mn n n n L W n n n square root: inverse of squaring 6 = 36 36 = 6 h 2 b Area=1/2bxh Important Notes d r negative numbers do not have square roots (in real number system) because the square of any real number is nonnegative

the principal square root of zero is zero, since 0 =0. That is 0=0

The principal square root of any number is positive

If c>0, then ( c ) =c 2 2 r Area= r^2 C=2 r V=4/3 r^3 SA=4 r^2 V=lxwxh SA=2lh+2wh+2lw V=lxwxh SA=2lh+2wh+2lw A Christmas tree ornament is in the shape of a semicircle on top of a triangle. How many square centimeters of copper is required to make the ornament if the height of the triangle is 6 cm. and the base is 4 cm. The amount of copper required equals the shaded area. This area is the sum of the area of the triangle and the semicircle. the triangle has a height of h=6 and base of b=4. The triangle has d=4 r=2 4 6 A=area of triangle+area of triangle

=1/2bh+1/2r^2

=1/2(4)(6)+1/22^2

=12+2

18.25cm^2

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