Question: How do we find the area of a circle?

Answer: That's very easy, the area of a circle is given by the formula A= πr2(squared) where A is the area and r is the radius. 

P.s. sometimes the radius isn't given to you, instead you are given the diameter. But dont worry because the diameter is just the radius times 2. So all you need to do to find the radius is divide the diameter by two! :)


NOW YOU TRY! :D


Find the area of the circle:

Question: How do we find the area of regular polygons?

Answer: Well you can divide a regular polygon into congruent isosceles triangles by drawing segments to each vertex, as shown in the following figure:

Regular Polygon Area Conjecture
The area of a regular polygon is given by the formulas: A=nas and A=Pa
                                                             2              2
Where A is the area, P is the Perimeter, a is the apothem, s is the length of sides, and n is the number of sides.
The apothem is the perpendicular bisector from the center of the polygon to the side.

As seen in the image above, to find the area all you have to do is follow either one of the formulas, right now i am going to use the formula: A=nas
                                                   2
So i will have to multiply 7(n) times 7(a) times 10(s) and divide it by 2. Thus, the area of the heptagon will be 245 square cm.


NOW YOU TRY! :D


Find the area of the following polygon:

Question: How do we find the area of parallelograms, kites, and trapezoids.

Answer: Woah that's a lot of shapes! :O
So lets start of with parallelograms :) and lets review what area is, area is the space inside a shape.

Okay so now, to find the area of a parallelogram you must multiply the base times the height. Its very similar to finding the area of a rectangle :) AREA=B*H

Okay lets move on into trapezoids:
To find the area of a trapezoid, you must add the two bases and divide them by two, then you multiply it by the height. It sounds a bit confusing so lets look at it visually :)

Finally. lets learn how to find the area of a kite:
To find the area of a kite you must multiply the two diagonals and divide them by two.

NOW YOU TRY! :D


Find the area of the following figure:

Question: How do we calculate the are of rectangles and triangles?

Answer: First of all what is area?
The area of a figure is the measure region enclosed by the figure.

Okay now that we know what area is, lets find out how to find the area of a rectangle! :D
The area of a rectangle can be found by multiplying the length(L) times the width(W). AREA=L*W

Figure 1

The area is measured in "units squared." If your rectangle is 5 inches long and 3 inches wide, to find the area you must multiply these two. so it would be: 5inches * 3 inches = 15inches(squared). Illustrated in Figure 1.

 Now lets find out how to find the area of a triangle. To find the area of a triangle you do one half the base times the height. AREA=(1/2)B*H

NOW YOU TRY! :D

Find the area of the rectangle above.

Question: How do we solve logic problems using conditionals?

Answer: I have already stated what a math statement is, but if you don't remember then i shall refresh your memory :)
A mathematical statement is a statement that can be judged to be true or false.
Ok now that we have gone over math statements, lets go over conditional.

A conditional falls under the category of logic in geometry. a conditional is a compound sentence created by putting together two sentences using the words "IF" and "THEN." A conditional is also made up of a hypothesis and a conlusion.

Example:
If I am laughing
then I am having fun
-The hypothesis of this conditional is "i am laughing" and the conclusion is "i am having fun."

NOW YOU TRY! :D


Form a conditional out of the following hypothesis and conclusion:
Hypothesis- it is raining
Conclusion- i will take my umbrella

Question: How do we find compound loci?

Answer: Ok so we already know what loci is, now what is a compound loci?
Well a compound locus problem involves two or more locus conditions in one problem. The way to differentiate a compound locus problem from a regular locus problem is by the words "AND" and "AND ALSO."

Example Problem:
A treasure is buried in your backyard.  The picture below shows your backyard which contains a stump, a teepee, and a tree.  The teepee is 8 feet from the stump and 18 feet from the tree.  The treasure is equidistant from the teepee and the tree AND ALSO 6 feet from the stump.  Locate all possible points of the buried treasure. (Figure 1)

Figure 1

Step 1:
Ok first read the problem and determine ONE of the locus conditions.
Step 2:
Solve the first locus solution and plot it.
Step 3:
Re-read the problem and identify the second locus condition.
Step 4:
Solve the second locus solution and plot it.

Ok so now lets see this would look like: (Figure 2)

Figure 2

Explanation of Solutions:
The red line represents the locus which is equidistant from the teepee and the tree (the perpendicular bisector of the segment AKA the first locus condition). The blue circle represents the locus which is 6 feet from the stump (the second locus condition). These two loci intersect in two locations. The treasure could be buried at either "X" location. 

NOW YOU TRY! :D


Two points A and B are 6 units apart.  How many points are there that are equidistant from both A and B and also 5 units from A?

1. 1
2. 2
3. 3
4. 4

Question: How do we find the locus of points?

Answer: So what is a locus? Well a locus is the set of all points that satisfy a given condition.

The plural of locus is loci

Ok now, let me explain how to identify the different types of locus problems.

There are 5 types of locus problems.

LOCUS PROBLEM #1: Finding the locus of one point

Figure 1

Whenever you are trying to find the locus of one point, the loci will be equidistantly away from the point. In figure 1, for example, the locus of points away from point P will be an equal distance away from the point, thus it will produce a circle with the original point (P) in the center.

LOCUS PROBLEM #2: Finding the locus of two points

Figure 2

Whenever you are trying to find the locus of two points, the loci will be a line through the middle of the two points. As illustrated in figure 2, for example, the locus of points away from points P and Q will be the perpendicular bisector of the line segment connecting the two points.

LOCUS PROBLEMS #3: Finding the locus of a line

Figure 3

Whenever you are trying to find the locus of a line, the loci will be two parallel lines on opposite sides of te original line. As illustrated in figure 3, for example, the locus of points of line L will be two lines, opposite sides, equidistant and parallel to the original line.

LOCUS PROBLEM #4: Finding the locus of two lines

Figure 4

Whenever you are trying to find the locus of two lines, the loci will be a line through the middle of the two lines. As illustrated in figure 4, for example, the locus of points of lines L1 and L2 will be another line halfway between both lines, and parallel to each other.

LOCUS PROBLEM #5: Finding the locus of two intersecting lines

Figure 5

Whenever you are trying to find the locus of two intersecting lines, the loci will be two intersecting lines halfway between the two original lines. As illustrated in figure 5, for example, the locus of points of lines L1 and L2 will be two additional lines that bisect the angles formed by the original lines.

NOW YOU TRY! :D

What shape would the locus of points 3 inches away from point A form?

1.A circle

2.A square

3.A triangle 

4.A rhombus 

Question: What is a mathematical statement?

Answer: A mathematical statement is a statement that can be judged to be true or false.
Ex. John Kennedy was a U.S. President.

Now lets go over negations.
Negations (not) always adds the opposite truth value to a mathematical statement.
Ex. John Kennedy was not a U.S. President.

When dealing with mathematical statements, we will usually look at more than one statement.
And since we will be dealing with multiple we must know what conjunctions are.
Conjunctions connect two statements with words such as and, but, or.

Ex. The President of the U.S. is Barack Obama and the vice president is Joe Biden.
And - both statements must be true for the statement to be true
Ex. I'm going to have pizza or tacos for dinner.
Or - Either statement (or both) can be true for the statement to be true.

Now let's talk about conditionals. The conditional is the most frequently used statement in the construction of an argument or in the study of mathematics (geometric proofs). There are four types of conditionals:
     -Conditional
               -Converse
                         -Inverse
                                   -Contrapositive

     
NOW YOU TRY! :D
Is the following a mathematical statement?

An owl is a fish. Yes or no?

Question: What is logic?

Answer: LOGIC IS THINKING!!!

Logic is simply a tool that we use to determine between true or false.

For example, we all use logic every single day.
Logic is kind of like common sense.

When you come to a busy street and you decide whether to cross or not, then you are using logic.

There is also inductive reasoning.
Inductive reasoning is when you use prior knowledge to predict the future (it might not always be true).

NOW YOU TRY! :D


How have you used logic today?