Journal 1

By: Esteban Lara

A point is a dot that describes the position of something somewhere.

A line is a straight connection of points that extends forever.

A plane is a flat surface that goes on forever on all directions.

Points, Lines and Planes

Point

N

Line Plane

Examples:

Collinear: Points that are on the same line are collinear.

Coplanar: Points that are on the same plane.

The similarities that they have are that they both have points that are on the same surface. A difference is that collinear have to be on the same line but coplanar don’t have to be aligned just on the same plane.

Collinear and Coplanar

Collinear Non collinear

Coplanar Non coplanar

Examples:

Line: A straight connection of points that extends forever.

Segment: A piece of a line with a beginning and end.

Ray: A line that goes on one direction only.

They are related because they all connect points that are collinear.

Line, Segment and Ray

Line Segment Ray

Examples:

An intersection is when two lines, rays or segments cross each other.

Examples:

Intersection

Postulate/Axiom: A postulate or axiom are the same thing just with different names. A postulate is a statement that is accepted as true without proof.

Theorem: A theorem is a statement that has been proven by previous postulates or another theorem.

The difference is that a theorem has already been proved and an axiom or postulate doesn’t necessarily have to be proven.

Postulate, Axiom and Theorem

Postulate Theorem

Examples:

The ruler postulate tells us the distance of a line. It tells us that every point in a line can be described by a real number. The two points are described as AB and the distance between them can be found by the subtraction of lA-Bl.

Ruler Postulate

Examples:

It’s used to find the distance between two points in a line. In other words you find the measure of a segment. The postulate is described by AB+BC=AC. Meaning the distance from point A to the midpoint and the midpoint to point C will give you the overall distance of the segment.

Segment Addition Postulate

Examples:

The distance between two points can be found by inserting the correct numbers on the distance formula.

d=

Distance Formula (finding the distance between two points)

Examples:

Congruence is measurements that might not know the value.

Equality is the same measurement and must know the value.

The shape is congruent if it has the same shape of another and same size and can be transformed in any way that doesn’t alter it’s shape but it’s equal if it’s exactly the same shape and size.

Congruence and Equality

Examples:Congruence

Equality

The sum of the areas of two squares on the sides A and B equals the area of the square C (hypotenuse). The formula is:

Pythagorean Theorem

Examples:

An angle is a formation of two rays that share a vertex (endpoint). Angles can be measured with a protractor.

The angle includes a vertex and at least two rays that have to form at least one corner.

Angles

Acute:Is the type of

angle that has a measurement smaller than 90 degrees.

Right:Is the angle

that has a measurement of exactly 90 degrees.

Obtuse:Is the type of

angle that has more than 90 degrees.

Types of Angles

Straight:Is an angle

that has a measure of 180 degrees.

Complementary:

Is when two angles share a vertex and a side. Their measure-ments add to 90 degrees.

Supplementary:

Is when two angles share a vertex and side but add up to 180 degrees.

More types of Angles

It states that if you add one adjacent angle to the other, it will give you the measurement of the whole angle. For example: If a point S lies in the interior of ∠PQR, then ∠PQS + ∠SQR = ∠PQR.

Angle Addition Postulate

Examples:According to angle addition postulate, m∠DAC + m∠CAB = m∠DAB. So, m∠DAB = 35° + 30° = 65°

The midpoint is a point in exactly the middle of a segment. It can be constructed by using a compass and marking to x’s on the top and on the bottom. To do this you need to set the compass to a little more than half the segment and use the same measurement on one side and on the other. You can also find it by using the formula:where P1(X1,Y1) andP2(X2,Y2) are inserted.

Midpoint

Examples:Find the midpoint between (–1, 2) and (3, –6).

Find the midpoint between (6.4, 3) and (–10.7, 4).

To bisect an angle you need to create a line with your compass on both rays. From that point where the ray intersects the compass-line you draw another line in front of the vertex, with the same measurement as before.

Example:

Angle Bisecting

Adjacent:Are two angles

that share a vertex and a side.

Linear Pairs:Are two angles

that form a straight line.

Vertical Angles:

Are angles that are non adjacent when two lines intersect, they’re congruent.

Adjacent, Linear Pairs and Vertical Angles

These angles are common because they both are angles that share a vertex and a line, the difference is that complementary angles add up to 90 degrees and supplementary add up to 180 degrees.

Examples:

Complementary Supplementary

Complementary and Supplementary angles

Square:To find the

perimeter you multiply by four the side and to find the area you square the length.

P=s+s+s+sor

P=4*sA=s^2

Rectangle:The perimeter is

the sum of the length multiplied by two and width multiplied by two. The area is length times width.

P=2L+2WA=l*w

Triangle:To find the

perimeter you add the sides and to find the area you multiply base times height then divide it by two.

P=s+s+sA=b*h/2

Perimeter and Areas

Examples for Square

Examples for Rectangle

Examples for Triangle

P=16in

To find the area of a circle you must multiply Pi times the radius (which is the distance from the center to any point of the circumference) squared. And to find the circumference (perimeter) you must multiply 2*Pi times the radius.

P= 2*Pi*r A= Pi*r^2

Circle Area and Circumference

Examples

Area=Pi*r^2Area=18.09cm^2Perimeter=2Pi*rPerimeter=15.08cm

The five steps are: first understand the problem; second, identify the variable you need; third, draw the problem; fourth, multiply or divide the numbers that have a variable to leave it alone; and last, solve.

Five Steps

I have a terrain that is 20 times 40 meters big. I would like to know the area of it. What is it?

1) The length is 40 and width 202) The variable is area3)

4) A=20*405) The area of my terrain is 800m^2

Example:

Translation:It’s when you

slide an object in any direction.

Rotation:It’s to rotate a

figure around a point.

Reflection:It’s to to

reflect your figure across any line.

Transformations

http://www.rmm.cl/usuarios/lfuentes01/imagen/t5.gif http://4.bp.blogspot.com/_RJ1GL9CAxSc/S5dosSaBXjI/AAAAAAAABkA/5BDM5OPGY4A/

s400/par_or7.gif http://image.wistatutor.com/content/feed/u342/plane.GIF http://www.onemathematicalcat.org/Math/Geometry_obj/graphics/collinear_points.pn

g http://www.icoachmath.com/Sitemap/images/Coplanar1.jpg http://img.youtube.com/vi/JAG1zsIt93U/0.jpg http://www.regentsprep.org/Regents/math/geometry/GG1/plane3.gif http://www.homeworkhelp.com/CMS/hwhlesson/mathematics/geometry/ch1/2/images

/010201_1.gif http://image.wistatutor.com/content/feed/tvcs/b_6.PNG http://strader.cehd.tamu.edu/Mathematics/Geometry/PolygonLesson/Details/example

1.gif http://www.teach-nology.com/worksheets/math/geometry/intersect/ver1/1.gif http://www.teach-nology.com/worksheets/math/geometry/intersect/ver1/2.gif http://content.tutorvista.com/maths/content/geometry/loci%20and%20concurrency%

20theorems/images/img23.gif http://4.bp.blogspot.com/_FXeaJy4hqRs/STMWdbARgSI/AAAAAAAAAFE/rf2i-rBg8BI/s40

0/crossroads.jpg

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