Question

Medal + Fan for anyone who helps!!! I could also raise your smart score as high as you would like!

Answer 1

Hi

Answer 2

So what part of the project do you need help with?

Answer 3

Hello?

Answer 4

Can I have my smart score to 100?

Answer 5

Story my tablet is doing weird things. I need help with everything and you can choose witch ever one you want. And yea

Answer 6

Ok Do you know what tessalations are?

Answer 7

No. If you could do this for me that would be great!

Answer 8

ok

Answer 9

For tessalations aka #2, At 2b. All you hav to do is get any type polygon and draw/color You may only chose the rectangle once though.

Answer 10

Problem is I dunno where to get one and ui just don't know how to do that

Answer 11

What do you mean U just draw a rectangle and another polygon and draw then color it

Answer 12

I not u sorry. That shy I'm saying if you could do that...

Answer 13

?

Answer 14

What

Answer 15

what do you mean by " I not u sorry. That shy I'm Saying if you could do that"

Answer 16

btw this should help you

Answer 17

Could you do that so can submitted it to my teacher? I will raise your smart score I promise

Answer 18

Geometry is often thought of as getting its start in Western culture in about 300 B.C., with the work of Euclid, a Greek mathematician. Euclid gathered together all of the mathematics that was known at the time and organized it into a collection called Elements. Euclid began by setting down the simplest mathematical assumptions or statements, which he called axioms. Then he reasoned that certain other statements, based on the axioms, must also be true. These statements he called theorems.

The system of using axioms as statements to be accepted without proof, then using the axioms to prove other statements, or theorems, was then used throughout all of mathematics and is still the way new mathematics is created.

In his Elements, Euclid also defined various geometric figures and described their basic properties, or characteristics. These figures included circles and other shapes made of straight or curved lines. Euclid also explained certain geometrical relationships and demonstrated how geometric figures can be constructed and measured. All of these ideas are still basic to an understanding of geometry, and they are the first things the student of geometry usually studies. This article will discuss the principal geometric figures.

Plane Figures

Euclid began his study of geometric figures with the plane. A plane is a flat surface like the top surface of a table, the outer side of a box, or a football field. We think of simple shapes, or figures, on the plane as enclosing part of the plane.

Any enclosed figure made from line segments is called a polygon. If the sides and the angles of the figure are all equal, it is called a regular polygon.

Angles
Angles are used in defining and describing many plane figures, so we will examine them before polygons. An angle is a wedge-shaped piece formed as an opening between two lines. Angles are measured with a protractor or an angle ruler, which are marked off in measures called degrees. The symbol ° is used for the word "degree."

Angles that make square corners and measure 90° are called right angles. If an angle measures less than a right angle, it is called an acute angle. If it measures more than a right angle, it is an obtuse angle.

Lines that meet at right angles are called perpendicular. If a pair of lines in a plane never meet, they are called parallel.

Triangles
The simplest polygon is the triangle, which is made of three line segments joined together. If all the line segments, or sides, are the same length, the figure is an equilateral triangle. All the angles in an equilateral triangle are also equal. If only two sides are the same length, the figure is an isosceles triangle. The two angles opposite the two equal sides of an isosceles triangle are also equal. Mathematicians working on computers often ask for a random triangle, one in which the lengths of the sides are unpredictable.

If one of the angles in a triangle is a right angle, the triangle is called a right triangle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs.

Triangles are widely used in science, navigation, and construction.

The Pythagorean Theorem
The Greek mathematician Pythagoras, who lived in the 500's B.C., worked out an important theorem about the relationships between the lengths of the legs and the length of the hypotenuse of a right triangle. Pythagoras first measured and then squared the lengths of the two legs of a right triangle. For example, if one leg measured 3 inches, he squared the 3 by multiplying it by itself, getting 3 × 3, or 9. If the other leg measured 4 inches, squaring the 4 produced 16. Pythagoras then added: 9 + 16 = 25. The sum, he discovered, was equal to the length of the hypotenuse squared. The length of the hypotenuse in this case would be 5 inches since 5 squared is 25.

Pythagoras' theorem can be stated this way: For any right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the length of the sides.

Quadrilaterals
A figure with four line segments enclosing a plane is called a quadrilateral. The word "quadrilateral" comes from a Latin word meaning "four-sided." One type of quadrilateral, the parallelogram, has opposite sides that are parallel and equal in length. When one of the angles in a parallelogram is a right angle, the figure is a rectangle. If all the sides of the parallelogram are the same length, the figure is a rhombus, and if one of the angles in a rhombus is a right angle, the figure is a square.

A less common quadrilateral is the trapezoid. It has only two parallel sides and they are unequal in length.

If a quadrilateral can be placed in a circle so that all four corners lie on the circle, it is called a cyclic quadrilateral. Squares and rectangles are cyclic quadrilaterals.

Other Polygons
There are many other polygons that are classified according to their number of sides. One example is the regular five-sided polygon called a pentagon, which was studied by the Greeks for the relationships that can be found in its sides and angles. Regular six-sided polygons called hexagons have been used in ancient and modern times to tile floors because they fit together so well. Today, the common stop sign takes the shape of the regular eight-sided polygon, which is the octagon.

Circles
A portion of a plane can be enclosed by figures other than polygons. For example, all of the points that are an equal distance from a fixed point describes the figure called the circle. Circles can be found everywhere. Wheels, many coins, and the moon in its full phase all have circular shapes.

A line through the center touching both sides of the circle is the diameter. Half of the diameter is the radius. The distance around a circle is the circumference.

To calculate the circumference of the circle, we multiply the diameter of the circle by a special number called pi. The symbol for pi is π and its value is approximately 3.14. If the diameter of the circle is 5 inches, for example, we multiply 5 × 3.14 to get a circumference of 15.7 inches, which is a little greater than three diameters.

Conic Sections
Menaechmus, a Greek mathematician who lived in the 300's B.C., is credited with the discovery of a group of curved plane figures that are formed by making slices through different sections of a cone. If you slice a cone straight across, the resulting section is a circle. If, instead, you slice a cone on an angle, you get an ellipse. Ellipses occur in the paths the planets take as they revolve around the sun, and in bridges, arches, and other constructions. If you increase the angle of the slice, you get a parabola, and increasing the angle even more yields a hyperbola. A baseball thrown into the air will travel in the path of a parabola.

Three-Dimensional Geometry

While most basic geometry that is studied is two-dimensional—concerned with figures having only length and width—we live in a three-dimensional world. From soccer balls and tin cans to Egyptian pyramids and barnyard silos, almost everything around us has three dimensions—length, width, and height.

Regular Polyhedra
Euclid ended his geometric studies with an exploration of the three-dimensional figures called solids that can be constructed from two-dimensional polygons. For example, if you join four equilateral triangles together at their corners, you will produce a solid called a tetrahedron. The four triangles make up the faces of the tetrahedron. The faces are joined at their six edges, and the edges end at four corners called vertices.

The tetrahedron is one of a group of solids called polyhedra. The singular of polyhedra is polyhedron. A polyhedron is a figure with flat faces and straight edges.

Two other polyhedra can be constructed from equilateral triangles: the octahedron, which has eight faces, and the icosahedron, with twenty faces.

One polyhedron, the cube, or hexahedron, has six faces and is built from squares. The dodecahedron is constructed from regular pentagons and has twelve faces.

These five figures make up a special group called the regular polyhedra, or the Platonic solids. All the faces in a regular polyhedron are the same size and shape. The Greeks thought that the five regular polyhedra reflected the universe and its four elements—air, water, fire, and earth.

Other Solid Forms
Polyhedra that are not regular include the prism and the pyramid. A prism has two parallel bases, or ends, and its other faces are parallelograms. A nonregular pyramid has one base that is not an equilateral triangle, and its other faces are triangles.

Not all solids are polyhedra with plane faces. Some solids include curved surfaces. The sphere is a solid with a curved surface. We see spheres in many types of balls used in sports and recreation. In a sphere, the distance from the center of the sphere to every point on its surface is the same.

The common can is a good model for another solid called a cylinder. The cone is probably most familiar to us in the form of the cone that holds scoops of ice cream.

Measuring Area and Volume

Measurement is an important part of geometry. From ancient times, geometers—mathematicians who specialize in geometry—have concerned themselves with measuring the areas and volumes of geometric figures.

Area
Area is the amount of surface covered by a figure. It can be measured by first covering a shape with standard unit squares, such as square inches or square centimeters, and then counting the number of standard squares in the shape. Imagine a rectangle divided into square centimeter units, with five columns and four rows. How many square centimeters do you get by counting all the unit squares? There are 20 unit squares, so the area of this rectangle is 20 square centimeters.

If the shape is an orderly shaped figure such as a square, or a rectangle, there is a quicker way of counting the unit squares. Just multiply the number of squares in each row (5) by the number of rows (4) to get 5 × 4 = 20 square centimeters. This shortcut method for finding the area of a rectangle can also be stated in a mathematical equation called a formula. Since there are 5 square centimeter units along the length of the shape and 4 square centimeter units along the width, the formula would then be written, Area = length × width, or A = l × w.

If you want to find the area of a triangle or some other shape in which the unit squares do not fit well, you could estimate the number of unit squares. Or, for an isosceles triangle, for instance, you could cut the triangle down the middle into two pieces and refit the pieces into a rectangle. Then find the number of unit squares in the rectangle. Similar methods work for other shapes.

Students sometimes find it useful to use a small plastic sheet with a grid of standard squares marked on it to measure irregular shapes.

Volume
The amount of space a solid figure occupies or fills is its volume. The volume of a figure is measured using methods similar to those used for finding areas. The unit of measure for volume is the standard unit cube such as a cubic inch or a cubic centimeter. We can pack a figure, a rectangular box, for example, with unit cubes and then count the number of cubes in the box. Or we can use the quicker method of counting the number of unit cubes in the bottom of the box, then count the number of layers of cubes necessary to fill the box, and then multiply the two numbers. If the bottom of the box had 6 rows of inch or centimeter cubes and it had 4 cubes in each row, there would be 6 × 4, or 24 inch or 24 centimeter cubes. If it then took 3 layers to fill the box to its height, we would multiply 24 × 3 to get a volume of 72 cubic inches or centimeters. Or, instead, we could use the formula Volume = length × width × height, or it may be written V = l × w × h.

To find the volume of odd-shaped or irregular objects, we could estimate the number of unit cubes in the object. Another method would be to pour water into the object and then pour the water into a special container marked in cubic units.

Similarity, Congruence, and Symmetry

Geometric figures that have the same shape but not necessarily the same size are called similar. In similar shapes, all corresponding angles are equal to each other. Whenever we build a scale model of an airplane or draw a map to scale, we are dealing with similar figures.

Two or more figures that have exactly the same size and exactly the same shape are called congruent. The tops of the student tables or desks in a classroom or the pages of a book are likely to be congruent to each other.

A figure is said to have symmetry if the parts of the figure on opposite sides of a point, line, or plane correspond exactly.

Transformation Geometry

Transformation geometry is concerned with those properties, or characteristics, of a figure that are not changed by certain rigid motions. Such motions are called isometries. The word "isometry" means "the same measure." An isometry, thus, is a movement of a figure to a new position, leaving all its measurements, including size and shape, the same.

One type of motion in transformation geometry is the flip, or line reflection. In a line reflection, every point in the figure is symmetrical to a corresponding point in the image on the other side of a line called the line of symmetry.

In the motion called a slide, or translation, a figure is moved up, down, to the left or right, or diagonally.

If a figure is rotated, or turned, around a fixed point, the motion is called a turn, or rotation.

Transformations can be used to create designs such as those made famous by the Dutch artist, M. C. Escher. These designs, consisting of interlocking shapes, are called tessellations, or tilings. Escher made extensive use of flips, slides, and turns to create his designs, rendered as drawings, woodcuts, or paintings.

Beyond the Third Dimension

The idea of dimension has long fascinated mathematicians. We can think about as many dimensions as we wish. Let us look at the simplest dimensional idea, which is the single point. We will say that the point represents zero dimensions.

Imagine the point shifting through space a short distance to generate a line segment with 2 endpoints. Counting only endpoints and segments, we have 2 points and 1 segment, representing 1 dimension.

Now imagine the segment shifting in a new dimension to generate a square, a two-dimensional object. We now have 4 endpoints, 4 segments, and 1 square.

Next, the square shifts perpendicular to itself to generate a cube with three dimensions. Now we have 8 points, 12 segments, 6 squares, and 1 cube.

We can put this data into a table to help us find some patterns. In one pattern, the number of points is doubled each time we shift the figure. That pattern is easy. Now look at what happened when the segment shifted, resulting in 4 points, 4 segments, and 1 square. The 4 points represent the 2 old points and 2 new points. The 4 segments represent the old segment, the new segment, and 2 new segments produced by the points that shifted.

Looking at the 6 squares that were generated in the shift from two to three dimensions, we can see that they were the old square, the new square, and 4 squares generated by the 4 shifting segments. In the table we can calculate the 6 by doubling the 1 directly above the 6 and adding the 4 to the left of the 1.

We can continue to extend the table by using the pattern of selecting a cell, or box, doubling the number above the cell, and adding the number to the upper left of the cell. To compute the numbers that would go in the row for the fourth dimension, we first double the 8 points, giving us 16 points. The number of segments would be 2 × 12, or 24, plus 8, or 32. The number of squares would be 2 × 6, or 12, plus 12, or 24. The number of cubes would be 2 × 1, or 2, plus 6, or 8. This produces a four-dimensional figure called a hypercube, or tesseract. With a computer we can produce an image of a hypercube, formed as the result of shifting a cube.

In the hypercube you can easily find and count the 16 points. It is a little harder to count the 32 segments. Some of the squares look strange because of your point of view, but it is possible to find 24 squares. Of the 8 cubes, you can easily see the large outer cube and the small inner cube. The other 6 cubes are nested on the 6 sides of the small cube and are bounded by the large cube.

You could extend the table to describe a five-dimensional or a six-dimensional cube. These are more than fanciful ideas. Ideas and figures of more than three dimensions are used in some multidimensional statistical tests and in other fields of mathematics.

Non-Euclidean Geometry

Ideas about non-Euclidean geometry, a geometry that is different from that of Euclid, originated with the great German mathematician Carl Friedrich Gauss in the 1800's. These ideas were subsequently worked out in detail by three other mathematicians—Nikolai Lobachevsky from Russia, János Bolyai from Hungary, and Bernhard Riemann, also from Germany.

One of Euclid's basic assumptions, the parallel postulate, had troubled mathematicians for years. It seemed to be more complicated than the other assumptions, and some mathematicians thought it should be classified as a theorem. The parallel postulate states that through any point not on a line, exactly one line can be drawn through the point parallel to the other line. Lobachevsky and Bolyai decided to find out what would happen if they made an alternative assumption to the parallel postulate. Their assumption stated that through any point not on a line, more than one line can be drawn through the point parallel to the other line. Riemann worked on another assumption, which stated that there were no parallel lines.

These ideas were developed into a new non-Euclidean geometry that forced mathematicians to change many former conceptions. The assumption had been that Euclidean geometry described our world very well. Non-Euclidean geometry, however, caused mathematicians and others to perceive the world in completely new and different ways and to develop new mathematical models with which to test their ideas. (More information about non-Euclidean geometry is in Mathematics, History of.)

Geometers continue to explore alternative geometries, looking for new ideas, patterns, and models. This contributes to our mathematical knowledge and to the development of major applications in science and technology.

Answer 19

What is that for?

Answer 20

To help u understand what tessalations are. I know it may say things about circles and things but If u read on then ull find out what tessaltions are.

Answer 21

For 2b all youy have to do is draw 2 polygons and color them

Answer 22

I'm so confused. Could you just do the assignment and send it on here so I can submitted it?

Answer 23

ok

Answer 24

@myininaya This guy want me to do a whole project for him plz help

Answer 25

So at your saying is... all I have to do is draw two angles and color them? That makes so sence

Answer 26

I didnt say that ... I said u draw two polygons

Answer 27

Oh wait you dont know what those are

Answer 28

Polygons are shapes like rectangles -.-

Answer 29

Okayha to I do next.