How Do We Find The Locus Of Points?

-A Locus is the set of all points that satisfies a given condition.

-A locus is a general graph of a given equation.

There are five basic locus theorems (rules).
Each theorem will be explained in detail in the following sections under this topic. Even though the theorems sound confusing, the concepts are easy to understand.

Locus Theorem 1:

The locus of points at a fixed distance, d, from point P is a circle with the given point P as its center and d as its radius.

Locus Theorem 2:

The locus of points at a fixed distance,d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.

Locus Theorem 3:

The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.

Locus Theorem 4:

The locus of points equidistant from two parallel lines, l₁ and l₂ , is a line parallel to both l₁ and l₂ and midway between them.

Locus Theorem 5:

The locus of points equidistant from two intersecting lines, l₁ and l₂, is a pair of bisectors that bisect the angles formed by l₁and l₂ .

Remember:
A locus is just a set of points.

You Now Know How To Find The Locus Of Points. Remember The Theorems, & Finding The Loci Of Points Will Be A Piece Of Cake ^.^

Source: Shnat's Notes & www.regentsprep.org/regents/math/geometry/GL1/What.htm

How Do We Review Logic?

Review Of Logic!

One of the goals of studying mathematics is to develop the ability to think critically. The study of critical thinking, or reasoning, is called logic.

1. "A square has 4 equal sides." What is the truth value of the negation of this sentence? Explanation Sentence: A square has 4 = sides. (T) Negation: A square does not have 4 = sides. (F) Answer False

2. What is the truth value of the statement "2 + 4 = 6 and 9 is a prime number."? Explanation 2+4=6 is true 9 is a prime number is false T and F is F Remember, BOTH must be T for AND to be T. Answer False

3. What is the truth value of the statement "Triangles have 3 sides or squares have 3 sides."? Answer True Under OR only one fact needs to be true.

4. What is the truth value of the statement "2 is even and 4 is odd."? Answer False For AND to be true, both facts must be true.

5. What is the truth value of the statement "If dogs bark, then horses quack."? Answer False -- T implies F is F.

Sentences, or statements, that have the same truth value are said to be logically equivalent. ("equivalent" means "equal")

In logic, a conjunction is a compound sentence formed by combining two sentences (or facts) using the word "and." A conjunction is true only when BOTH sentences (or facts) are true.

In logic, a conditional is a compound statement formed by combining two sentences (or facts) using the words "if ... then." A conditional can also be called an implication.

Source: RegentsPrep.Org

How Do We Solve Logic Problems Using Conditionals?

In This Post You Will Learn What A Conditional Is & How To Solve Logic Using It.

In logic, a conditional is a compound statement formed by combining two sentences (or facts) using the words "if ... then."

There Are Three Types Of Conditionals & One Special Conditional:

The converse of a conditional statement is formed byinterchanging the hypothesis and conclusion of the original statement. In other words, the parts of the sentence change places. The words "if" and "then" do not move.

Example:

Conditional: "If the space shuttle was launched, then a cloud of smoke was seen."

Converse: "If a cloud of smoke was seen, then the space shuttle was launched."

HINT: Try to associate the logical CONVERSE with Converse™ sneakers -- think of the two parts of the sentence "putting on their sneakers" and "running" to their new positions.

** It is important to remember that the converse does NOT necessarily have the same truth value as the original conditional statement.

The inverse of a conditional statement is formed bynegating the hypothesis and negating the conclusion of the original statement. In other words, the word "not" is added to both parts of the sentence.

Example:

Conditional: "If you grew up in Alaska, then you have seen snow."

Inverse: "If you did not grow up in Alaska, then you have not seen snow."

HINT: Remember that to create an INverse, you will need to INsert the word NOT into both portions of the sentence. Since you are actually negating each part of the sentence, you may also use other words (in addition to NOT) to create the negation.

** It is important to remember that the inverse does NOT necessarily have the same truth value as the original conditional statement.

The contrapositive of a conditional statement is formed bynegating both the hypothesis and the conclusion, and then interchangingthe resulting negations. In other words, the contrapositive negates and switches the parts of the sentence. It does BOTH the jobs of the INVERSE and the CONVERSE.

Example:

Conditional: "If 9 is an odd number, then 9 is divisible by 2." (true) (false)	This statement is logically FALSE.
Contrapositive: "If 9 is not divisible by 2, then 9 is not an odd number." (true) (false)	This statement is logically FALSE.

HINT: Remember that the contrapositive (a big long word) is really the combining together of the strategies of two other words: converse and inverse.

**An important fact to remember about the contrapositive, is that it always has the SAME truth value as the original conditional statements.

In logic, a biconditional is a compound statement formed by combining two conditionals under "and." Biconditionals are true when bothstatements (facts) have the exact same truth value.

A biconditional is read as "[some fact] if and only if [another fact]" and is true when
the truth values of both facts are exactly the same
-- BOTH TRUE or BOTH FALSE.

Great You Now Know The Three Different Types Of Conditionals! Create An Inverse, Converse, Bioconditional & Contropositive For The Following Conditional:

"If Mary is eating the pizza, then Mary is hungry."

~

Sources:

http://regentsprep.org/Regents/math/geometry/GP2/Lconvers.htm

http://regentsprep.org/regents/math/geometry/GP2/Lcontrap.htm.

http://regentsprep.org/Regents/math/geometry/GP2/Linvers.htm

What Is A Mathematical Statement & What Is Logic?

In This Post, You Will Learn What A Mathematical Statement Is, Types Od Mathematical Statements, & What Logic Is.

*One of the goals of studying mathematics is to develop the ability to think critically. The study of critical thinking, or reasoning, is called logic.

*All reasoning is based on the ways we put sentences together. A mathematical sentence is a sentence that states a fact & can be judged to be true or false.

-Questions and phrases are not mathematical sentences since they cannot be judged to be true or false.

Ex:"An isosceles triangle has two congruent sides." is a true mathematical sentence.

There are two types of mathematical sentences:

An open sentence is a sentence which contains a variable.

• "x + 2 = 8" is an open sentence -- the variable is "x."
• "It is my favorite color." is an open sentence-- the variable is "It."

• A closed sentence, or statement, is a mathematical sentence which can be judged to be true or false. A closed sentence, or statement, has no variables.

• "Garfield is a cartoon character." is a true closed sentence, or statement.
• "A pentagon has exactly 4 sides." is a false closed sentence, or statement.
• 
  

In logic, a conjunction is a compound sentence formed by combining two sentences (or facts) using the word "and." A conjunction is true only when BOTH sentences (or facts) are true.

Examples:

- "Blue is a color and 7 + 3 = 10." (T and T = T)
Since both facts are true, the entire sentence is true.

-"One hour = exactly 55 minutes and one minute = exactly 60 seconds."
(F and T = F) Since the first fact is false, the entire sentence is false.

And lastly, Sentences, or statements, that have the same truth value are logically equivalent.

Great! Now Identify Which Problems Are Mathemathical Statements & Whether They Are True Or False:

1. 7+8.

2. A square has the same amount of sides as a circle.

3. All triangles have angle relationships.

Sources: http://regentsprep.org/regents/math/geometry/GP1/sentence.htm

http://regentsprep.org/regents/math/geometry/GP1/conjunct.htm

How Do We Use The Other Definitions Of Transformations?

Isometry - length is preserved - the
figures are congruent.

Direct Isometry - orientation is preserved - the order of the lettering in the
figure and the image are the same, either both clockwise or both
counterclockwise.

Opposite Isometry - orientation is not preserved - the order of the lettering is
reversed, either clockwise becomes counterclockwise or counterclockwise
becomes clockwise.

Orientation- Refers to the arrangement of points relative to the one another, after a transformation has occured.

Line Reflection (Opposite Isometry)

-Keeps Distance, Angle Measure, & Parrellism.

-Reverse Orientaton

Point Reflection (Direct Isometry)

-Keeps Distance, Angle Measure & Parrellism.

-Same Orientation

Translations(Direct Isometry)

-Keeps Distance, Angle Measure & Parrellism.

-Same Orientation.

Rotations (Direct Isometry)

-Keeps Distance, ANgle Measure & Parrellism.

-Same Orientation.

Diolation(Not Isometry)

-Keeps Angle Meauser & Parrellism.

-Lengths Not Same.

Great! Now Solve The Following Problem:

Which of the following transformations creates a figure that is similar (but not congruent) to the original figure? I. translation II. rotation III. dilation

[1] I only [2] II only [3] III only [4] II and III

How Do We Graph Rotations?

-A rotation is a transformation that turns a figure about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. (notation R_degrees )

-An object and its rotation are the same shape and size, but the figures maybe turned in different directions.

Rotation of 90°:

Rotation of 180°:

Rotation of 270°:

-When working in the coordinate plane, assume the center of rotation to be the origin unless told otherwise. A positive angle of rotation turns the figure counterclockwise, and a negative angleof rotation turns the figure in a clockwise direction.

Congratulations! You now know how to graph rotations. You can now try the following problem?

Rotate the triangle 90 degrees.

How Do We Graph Dilations?

-A dilation is a transformation (notation ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.

-The description of a dilation includes the scale factor.

*If the scale factor is > than 1, than the image will stretch.

*If the scale factor is > than 0 and < than 1, the image will shrink.

-To find the dimensions of the dilated image, multiply the dimensions of the original image by the scale factor.

PROBLEM: Draw the dilation image of pentagonABCDE with the center of dilation at the origin and a scale factor of 1/3. OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3). HINT: Multiplying by 1/3 is the same as dividing by 3! -Congratulations! You now know how to graph dilations. You can now try solving the following problem: What are the coordinates of the point (2,-4) under the dilation D(-2)?