basic geometry review

7/28/2019 Basic Geometry Review

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6/17/2013 Ventura College Mathematics Department 1

Basic Geometry Review


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Undefined Geometric Terms

PointA

Line

PlaneABC

AB


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This is rayAB

PointAis the vertex or

endpoint ofAB; always write

the name of the endpoint first

Definition:AB is the set of all

points ConAB such thatA is

not strictly between B and C

Half-lines (Rays)

AB

AB

AB

AB


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Line Segments

The green portion

is line segmentAB

PointsA & Bare endpoints;

distance between them isAB

Definition:AB is bounded by

endpointsA and B; it contains

every point onAB that is

between endpointsA and B

AB

AB

AB


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Circles (1 of 5)

Definition: A circle is

the set of all points

lying in a plane at a

fixed distance r(theradius) from a given point (the

center of the circle)

A diameter dis any line segment

whose endpoints lie on the circle,and which passes through

(contains) the center of the circle


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Circles (2 of 5)

A secant line is anyline that touches thecircle at exactly twopoints

A tangent line is anyline that touches the circle atexactly one point

A chord is any line segmentwhose endpoints lie on the circle,but which does not pass throughthe exact center of the circle


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Circles (3 of 5)

The circumference isthe full outer edge of thecircle, or the length of it

An arc is any continuousportion of the circumference

A sector is the wedge-like shapebounded by two radii and the arc

that lies between them A segment is the shape formed

by a chord and the arc thatextends between its endpoints


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Circles (4 of 5)

Formulas:

Diameter: d= 2r

Circumference: C= 2r= d

Area: A = r2

Pi: 3.14159265358979


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Circles (5 of 5)

Equations and unit circles

The equation of a circle whose

center is located at the origin of a

Cartesian coordinate system isx2 + y2 = r2

A unit circle is a circle that has a

radius of one unit (r= 1)

So the equation of a unitcirclewhose center is located at the origin

of a Cartesian coordinate system is

x2 + y2 = 1


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Angles

An angle BAC(orCABorA, if the shorter nameis clear) is the figure formed whentwo rays (the sides or legs ofthe angle) share a single endpoint

A(the vertex of the angle); thevertex is always the middle letter

Latin or Greek lowercase letters,such as a, b, , , , or, are alsoused to name angles intrigonometry and higher math


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Angle Measure (1 of 3)

Pac-Mans jaw forms

an angle (the black

wedge in the figure);

the measure of the angle is anumber that tells us about the

size of the wedge (how far open

Pac-Mans jaw has become)

The angles measure increasesas Pac-Man opens up wider


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One unit often used to measure

angles is the degree (symbol: )

Visit this web page* to learn about

different kinds of angles: Acute angles (measure m < 90)

Right angles (m = 90)

Obtuse angles (90 < m < 180)

Straight angles (m = 180)

Reflex angles (180 < m < 360)____________

* http://www.mathopenref.com/angle.html
http://www.mathopenref.com/angle.htmlhttp://www.mathopenref.com/angle.html


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If line segments, rays, orlines cross at a right angle(perpendicular), then asmall square is oftenadded to indicate this

Two angles whosemeasures add up to 90are complementary

Two angles whose measuresadd up to 180 aresupplementary


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Polygons (1 of 5)

The intuitive polygon: Draw a random assortment

of 3 or more points in a plane

Connect them so that each

point is the endpoint ofexactly two line segments,and no point lies on a givenline segment unless it is oneof that segments twoendpoints

The result is a polygon(some examples are shownin the figure at right)


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Polygons (2 of 5)

The strictly defined polygon(you wont be tested onthis): A polygon is aclosed path composed of afinite sequence of straightline segments

Other terms (you may betested on these): The line segments are called

sides of the polygon

Each corner is called avertex of the polygon


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Polygons (3 of 5)

Polygons are what anaverage person would callshapes but there aresome restrictions: Polygons have no curvy

parts; the definition (see theprevious slide) requires eachside to be straight

So, although circles, ellipses,parabolas, and other curvythings are called shapesalso, they are notpolygons


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Polygons (4 of 5)

Mathematicians classifypolygons by the number ofsides (or vertices) theyhave; the names usedhave mostly Greek roots: 3 sides = triangle or trigon

4 sides = quadrilateral ortetragon

5 sides = pentagon

6 sides = hexagon

8 sides = octagon, etc.


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Polygons (5 of 5)

Some polygons possesssymmetry; terms used forcertain types of symmetryinclude: Equiangular: All the vertex

angles have equal measures

Cyclic: All the vertices lieon a circle

Equilateral: All the sides, oredges, have the same length

Regular: The polygon isboth cyclic and equilateral


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Triangle Properties (1 of 2)

The measures of the three vertex

angles always add up to 180

An equilateral triangle is always

equiangular (and vice-versa); ifeither of these is true, then both

are true, and the measure of each

vertex angle is exactly 60

An equilateral triangle is the only

kind of triangle that is regular


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Triangle Properties (2 of 2)

If the lengths of at least two

sides of a triangle are equal,

then it is called an isosceles

triangle

If all three sides of a triangle

have different lengths, then it

is called a scalene triangle


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Right Triangles

If one vertex angle of a triangle isa right angle (has a measure of90), then the triangle is a righttriangle, having these properties: The two remaining vertex angles are

automatically complementary

It may be either scalene orisosceles; if it is isosceles, then the

two remaining vertex angles bothhave equal measures of exactly 45

The Pythagorean theorem (AppendixA) relates the lengths of the 3 sides


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Quadrilateral Properties (1 of 2)

The measures of the four vertexangles always add up to 360

An equilateral quadrilateral is

called a rhombus; it is notnecessarily equiangular or square

An equiangular quadrilateral iscalled a rectangle; it is not

necessarily equilateral All four vertices of a rectangle areright angles, and therefore havemeasures of 90


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Quadrilateral Properties (2 of 2)

A square is a quadrilateral

that is both equilateral and

equiangular

A square is the only kind of

quadrilateral that is regular


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Appendix A:

Pythagorean Theorem Ifcis the length of the

hypotenuse (longest side),

and a and b are the lengths of the

legs (shorter sides), thena 2 + b 2 = c2

The hypotenuse is always the side

that does nottouch the right angle

The figure depicts a scalene triangle;some right triangles might also be

isosceles, but they can never be

equilateral


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Appendix B:

The Greek Alphabet alpha ()

beta ()

gamma ()

delta ()

epsilon () zeta ()

eta ()

, theta ()

iota () kappa ()

lambda ()

mu ()

nu ()

xi ()

omicron ()

pi ()

rho () , sigma ()

tau ()

upsilon ()

, phi () chi ()

psi ()

, omega ()

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