Geometry Notes — Isosceles Triangles

Isosceles triangle

Isosceles triangle – a triangle with at least two sides congruent

Legs of an isosceles triangle – the congruent sides of an isosceles triangle.

Base of an isosceles triangle – the side that is not one of the two congruent sides in the isosceles triangle

Base angle of an isosceles triangle – one of the two angles formed by a leg and the base in an isosceles triangle

Vertex angle of an isosceles triangle – the angle formed by the legs of an isosceles triangle

Theorem:  Isosceles Triangle Theorem — The base angles of an isosceles triangle are congruent.

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

The base angles of an isosceles triangle are always congruent. In the figures, the angles ∠ABC and ∠ACB are always the same.

Theorem:  The Converse of the Isosceles Triangle Theorem — If two angles of a triangle are congruent, then the sides opposite them are congruent.

The legs of an isosceles triangle are always congruent. In the figures, the sides AB and AC are always the same.

If all three sides are the same length it is called an equilateral triangle. Obviously all equilateral triangles also have all the properties of an isosceles triangle.

Geometry Notes — Triangles (art)

I just love geometry art. Outstanding.

Pre-Calculus Assignment — Book 2.1 Problem 91 Slides

Pre-Calculus Notes — Line of Best Fit

You got it, you might as well learn to do something on it than play games.

We are going to create a scatter plot graph.

First, turn on the Stat Plot feature. Press 2nd STAT PLOT

Let’s enter our data. Press STAT. Enter your data. Use L1 for you independent variable and L2 as your dependent variable. Be sure to QUIT the list when finished.

To get an appropriate Scatter Plot Graph, press ZOOM. 9:ZoomStat. Isn’t that nice.

To find the Line of Best Fit. Press STAT. CALC. 4:LinReg (ax+b).

a = the slope of the line. b = the y intercept of the line. Let’s graph this Line of Best Fit by entering the slope and y intercept.

Let’s graph this Line of Best Fit another way. Press Y=. We can enter the information by hand, or we can let the calculator import the slope and y-intercept.

To do this. On the Y= screen, press VARS. 5:Statistics. EQ. 1:RegEQ. And the information is imported into Y1=. Press GRAPH. Nice, huh?

So, let’s evaluate the function for 21,500.

We obtain $20,408.39. Nice.

We can do something a lot fancier with the calculator. Since we have the function graphed, here’s how.

Press VARS. Y-VARS. 1:Function. 1:Y1. Y1 because that is the name of the graph.

Geometry Notes — Exterior Angles of a Triangle

Exterior angle – an angle created by extending one side of a figure

Theorem:  The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

…..

In the figures above the exterior angle ∠ABP is equal to the sum of the remote interior angles ∠BAC and ∠ACB.

.

Proof:

………

.

.

If the equivalent angle is taken at each vertex, the exterior angles always add to 360°

Since the measure of an exterior angle equals the sum of its two remote interior angles, the exterior angle is greater than the measure of either individual remote interior angle.

Theorem:  The measure of an exterior angle of a triangle is greater than either of its two remote interior angles.

Statistics Notes — What Kind of Probability is this Example?

Link

The graduate program in orthodontics at the University of Tennessee begins in August of each year and continues for 34 consecutive months until graduation. This period of study provides for intensive instruction and training in the biological and clinical sciences related to the specialty of orthodontics. The course of instruction is designed to satisfy all requirements for eligibility for licensure as a specialist, the specialty board in orthodontics, and the Master of Science degree. The program, fully accredited by the Commission on Dental Accreditation, is well rounded and provides balanced training in clinical orthodontics, the basic sciences, and research. Our goal is to prepare the graduate to pursue many career possibilities: a clinical practice limited to orthodontics, a researcher or a teacher of orthodontics.

Admission Decisions

The probability of success in graduate study is assessed using many criteria, but important characteristics are:

1. Dental and undergraduate school achievement.
2. Performance on standardized tests.
3. Extracurricular activities
4. The interview.
5. Experience.
6. Recommendations

Minimally, to be considered for full-standing as a graduate student, an applicant must have earned a D.D.S. or equivalent degree awarded by an accredited institution, compiled at least a “B” coursework average, and rank in the upper 1/3 of his or her graduating class.

Statistics Assignment — Book Chapter 3

Pre-Calculus Notes — The Vertical Line Test

The vertical line test is a way to determine whether a relation is a function. The vertical line test simply states that if a vertical line intersects the relation’s graph in more than one place, then the relation is a NOT a function.

For a relation or graph to be a function, it can have at most a single y-value for each x-value. Thus, a vertical line drawn at any x-position on the graph of a function will intersect the graph at most once.

To use the vertical line test graphically, take a ruler or other “vertical line” and move it from one end of the x axis to the other. If the graph intersects the ruler or vertical line more than once at any given point, the graph is not a function. If the graph intersects the vertical line only once at any given point, it is a function.

These relations pass the vertical line test and are functions because of it.

.

.

These relations fail the vertical line test and are not functions because of it.

fail

.

fail

Pre-Calculus Notes — Domain and Range

In mathematics, the domain of a given function is the set of “input” values for which the function is defined.
The domain of a linear function would be all real numbers.
The domain of a quadratic function would be all real numbers.
The domain of any polynomial function would be all real numbers.
The domain of a square root function would only be the numbers that would allow the expression within the square root to remain greater than or equal to 0.
The domain of a rational function would need to omit any values that would cause any denominator to become zero.
The domain of a rational function with a square root in the denominator would be the numbers that would allow the expression within the square root to remain greater than 0.

In mathematics, the range of a function is the set of all “output” values produced by that function.

Pre-Calculus Notes — Relations and Functions

A relation is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned one or more elements of the range.

A relation is a diagram, equation, or list that defines a specific relationship between groups of elements.

Mathematical relations are basically ordered pairs. The expressions that we have graphed are all relations. The simple ones such as lines and circles, where we have graphed them based on our knowledge of the standard graphing form of those equations to the graphs we have drawn by first plotting points — these have all been relations.

.

A function is a correspondence between two sets (called the domain and the range) such that to each element of the domain, there is assigned exactly one element of the range.

A function is a relation whose every input corresponds with a single output. And that is the thing — Each input –> a single output.

Functions are special relations. What distinguishes a function from a relation is that each x value in a function has one and only one y value. You have done the function problem in the past that had the familiar function notation f(x). These were all functions. When you substituted a x value into the expression, you obtained a y value, and only one y value. This makes it a function. Some of the graphs that we have worked though this year are not functions, they are relations only. Your lines — functions. Your circles — relations, not functions. Your parabolas of x² — functions. Your parabolas of y² — relations, not functions.

Most functions are written using function notation. For example, the function y = x². You know that y is a function of x because for every number x you plug into x², you can get only one corresponding output. Written in function notation, that function looks like f(x) = x².

Function notation is handy for two reasons:

It contains the name of the function
It’s easy to tell the value you’re plugging into the function

.

Evaluate the function f(x) = 2x² + x – 3 for x = -1. Evaluating the function at x = -1 is the same as finding the value f(-1). Plug in -1 everywhere you see an x:

f(x) = 2x² + x – 3
f(-1) = 2(-1)² + (-1) – 3 = 2 – 1 – 3 = -2

So you have the ordered pair (-1,-2). You can plot this along with other point derived from this function to get a visual representationt of the function. Or you can look at it and remark at how pretty it is. Pretty.

.

The domain of a function is the x values from your ordered pairs. The domain of a function is the x values that you would plug into the expression and determine the corresponding y value. The domain of a function is the x values that are independent. The domain of a function is the x values and is sometimes called the argument of the function.

The range of a function is the y values from your ordered pairs. The domain of a function is the y values that you would determine or solve for after you plugged in a x value. The range of a function is the y values that are dependent on the x values. The range of a function is the y values and is sometimes called the image of the function.

.

What makes a relation not a function — In a y² parabola, such as x = y² – 4, all but one x value would result in two y values.

For instance, replace x with 0.

x = y² – 4
0 = y² – 4
4 = y²
y = 2 or -2

Which would produce the ordered pairs (0,2) and (0,-2). This is violation of a function requirement.

For a relation to be a function, there cannot be any operation or means that would permit a x value to paired with more than one y value.

.

.
…..As a mathematical term, “function” was coined by Gottfried Leibniz to describe a quantity related to a curve.
The word function was later used by Leonhard Euler during the mid-18th century to describe an expression or formula involving various arguments.