GEOMETRY1

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Karen Jo CruzAileen Grace Delima

Scheherazaide Yoradyl PahmMae Paradela????????????

________________________1A paper done in partial fulfilment of the requirements in MST 3 ?????????????

SIMILARITY

DefinitionTwo geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly “stretching,” possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called “AAA” condition. However, since the sum of the interior angles in a triangle is fixed, as long as two angles are the same, all three are, called “AA.”

Similar PolygonsSimilar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional. Example:

EXAMPLEProblem: Find the value of x, y and the measure of angle P.

Solution: To find the value of x and y, write proportions involving corresponding sides. Then use cross products to solve.4 x 4 7– = – – = –6 9 6 y

6x = 36 4y = 42

x = 6 y = 10.5

To find angle P, note that angle P and angle S are corresponding angles. By definition of similar polygons, angle P = angle S 86°.

Similar TrianglesIf triangle ABC is similar to triangle DEF, then this relation can be denoted as

In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°.

Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following:

This idea can be extended to similar polygons with any number of sides. That is, given any two similar polygons, the corresponding sides are proportional.

Special Similarity Rules for TrianglesThe triangle has a couple of special rules dealing with siilarity. They are outlined below:

Angle-Angle Similarity – if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

EXAMPLEProblem: Prove triangle ABE is similar to triangle CDE.

Solution: Angle A and angle C are congruent. Angle AEB and angle CED are congruent because vertical angles are congruent. Triangle ABE and triangle CDE are similar by Angle-Angle.

Side-Side-Side Similarity – if all pairs of corresponding sides of two triangles are proportional, then they are similar.

Side-Angle-Side Similarity – if one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional, then the two triangles are similar.

EXAMPLEProblem: Are the triangles shown in the figure similar?

Solution: Find the ratios of the corresponding sides. UV 9 3 VW 15 3––– = ––– = – ––– = ––– = – KL 12 4 LM 20 4

The sides include angle V and angle L are proportional. Angle V and angle L are proportional.Angle V and angle L are congruent.Triangle UVS and triangle KLM are similar by Side-Angle-Side.

Parallel Lines and TrianglesWhat do parallel lines and triangles have to do with similar polygons?  Well, you can create similar triangles by drawing a segment parallel to one side of a triangle in the triangle.   This is useful when you have to find the value of a triangle's side (or, in a really scary case, only part of the value of a side).

The theorem that lets us do that says if a segment is parallel to one side of a triangle and intersects the other sides in two points, then the triangle formed is similar to the original triangle.  Also, when you put a parallel line in a triangle, as the theorem above describes, the sides are divided proportionally.

EXAMPLEProblem: Find PT and PR.

Solution: 4 x because the sides are divided proportionally when you draw a – = – parallel line to another side 7 12

7x = 48 cross products

x = 48/7

PT = 48/7

PR = 12 + 48/7 = 132/7

TRIGONOMETRY

Trigonometry is among the earliest applications of Euclidean geometry.

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles.

The two branches of trigonometry are:

Plane trigonometry - which deals with figures lying wholly in a single plane

Spherical trigonometry - which deals with triangles that are sections of the surface of a sphere

It was invented in order to deal with problems in astronomy and has many other uses to the measurement of distance that are difficult or impossible to measure directly.

More recent developments in the study of trigonometric functions enable us to consider problems involving periodic phenomena such as sound waves.

The trigonometric functions of angles are the ratios of the various sides of a triangle.

Hypotenuse: The side opposite to the right angle in a triangle is called the hypotenuse. Here the side AC is the hypotenuse.

Opposite Side: The side opposite to the angle in consideration is called the opposite side. So, if we are considering angle A, then the opposite side is CB.

Base: The third side of the triangle, which is one of the arms of the angle under consideration, is called the base. If A is the angle under consideration, then the side AB is the base.

The theory of similar triangles will serve as a basis for introducing these definitions.

B1 B2 B3 B

C C1 C2 C3 A

The consequent proportions of corresponding sides can be rewritten as follows:

BC = B1C1 = B2C2 = B3C3 = s (1)AB AB1 AB3 AB3

AC = A1C1 = A2C2 = A3C3 = c (2)AB AB1 AB2 AB3

BC = B1C1 = B2C2 = B3C3 = t (3)AC AC1 AC2 AC3

Now, since angle A is common to all triangles, we can describe the ratios s, c, t with reference to the acute angle A:

s= opposite leg (to A) (1) hypotenuse

c= adjacent leg (to A) (2) hypotenuse

t= opposite leg (to A) (3) adjacent leg (to A)

This means that for the same acute angle A on any right triangle the ratios (1), (2), (3) are constant, hence we can give them names:

Define: sine A = opposite leg sin A hypotenuse

cosine A = adjacent leg abbreviated cos A hypotenuse

tangent A = opposite leg tan A adjacent leg

These are the three basic trigonometric functions of angle A.

Observe that angle A and angle B are complementary, i.e., A = B = 90 ° and that

sin A = cos Bcos A = sin B

We state the property above as follows: If two angles are complementary, then the sine of one equals the cosine of the other. This is known as the complementary angle property.

EXAMPLE

Problem: Find the values of sin A, cos A, and tan A.

Solution: sin(A) = opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548

Or simply:

sin(A) = 0.5548

Using the above measured triangle, this would mean that:

cos(A) = adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322

Or simply:

cos(A) = 0.8322

tan(A) = opposite / adjacent

= 4.00 cm / 6.00 cm= 0.6667

Or simply:

tan(A) = 0.6667

The angle A in the above triangle is actually very close to 33.7 degrees. So, we would say:

0.5548 = sin(33.7°)0.8322 = cos(33.7°)0.6667 = tan(33.7°)

Here is an easy way to remember these relationships for trig functions and the right triangle.

SOH - CAH - TOAIt is pronounced "so - ka - toe - ah".The SOH stands for "Sine of an angle is Opposite over Hypotenuse."The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."The TOA stands for "Tangent of an angle is Opposite over Adjacent."

TRIVIAFourier Ears Only!Writing a function as a sum of sines and cosines is called a Fourier series. In fact, your ears do Fourier series automatically! There are little hairs (cilia) in you ears which vibrate at specific (and different) frequencies. When a wave enters your ear, the cilia will vibrate if the wavefunction "contains" any component of the correponding frequency! Because of this, you can distinguish sounds of various pitches!

PERIMETER

The perimeter is the distance around a given two-dimensional object. The word perimeter is a Greek root meaning measure around, or literally "around measure". A polygon is 2-dimensional; however, perimeter is 1-dimensional and is measured in linear units. To help us make this distinction, look at our picture of a rectangular backyard. The yard is 2-dimensional: it has a length and a width. The amount of fence needed to enclose the backyard (perimeter) is 1-dimensional. The perimeter of this yard is the distance around the outside of the yard.

The perimeter of a polygon can always be calculated by adding all the length of the sides together. So, the formula for triangles is

P = a + b + c, where a, b and c stand for each side of it.

For quadrilaterals, the equation for the perimeter is: P = a + b + c + d.

For equilateral polygons, the equation for perimeter is: P = na, where n is the number of sides and a is the side length.

For circles the circumference is a kind of perimeter. The circumference is the distance around a closed curve. The equation for circumference is:

or

P stands for the perimeter, r stands for the radius π is the mathematical constant pi (π = 3.14159265...) d stands for the circle's diameter (twice the radius of a circle)

(The dot means multiply or times)

EXAMPLES

Problem 1: Find the perimeter of a triangle with sides measuring 5 centimeters, 9 centimeters and 11 centimeters.

Solution: P = 5 cm + 9 cm + 11 cm = 25 cm

Problem 2: A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the perimeter.

Solution 1: P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm Solution 2: P = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm

Problem 3: Find the perimeter of a square with each side measuring 2 inches. Solution 1: P = 2 in + 2 in + 2 in + 2 in = 8 in Solution 2: This regular polygon has 4 sides, each with a length of 2 inches. Thus we get:

P = 4(2 in) = 8 in

Problem 4: Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.

Solution 1: P = 4 cm + 4 cm + 4 cm = 12 cm Solution 2: This regular polygon has 3 sides, each with a length of 4 centimeters. Thus we

get: P = 3(4 cm) = 12 cm

Problem 5: The perimeter of a regular hexagon is 18 centimeters. How long is one side? Solution: P = 18 cm Let x represent the length of one side. A regular hexagon has 6 sides, so we can divide the perimeter by 6 to get the length of one side (x).

x = 18 cm ÷ 6 x = 3 cm

Problem 6: What is the circumference of a circle having a diameter of 7.9 cm, to the nearest tenth of a cm?

Solution: Using an approximation of 3.14159 for , and the fact that the circumference of a circle is times the diameter of the circle, the circumference of the circle is C = Pi × 7.9

=3.14159 × 7.9 = 24.81…cm, which equals 24.8 cm when rounded to the nearest tenth of a cm.

R

NON-EUCLEDIAN GEOMETRY

“One geometry cannot be more true than another; it can only be more convenient”.Poincaré

A non-Euclidian geometry is described as a geometry satisfying a system of axioms in which one or more are contrary to Euclid’s, particularly his fifth postulate P5.

Hyperbolic GeometryOne model for hyperbolic geometry is the disc model H which makes use of a system of circles of a Euclidian plane. It is based on the work of Henry Poincare (1854-1912) and as described as:

C C C C

Q

P S T S

(a) (b) (c) (d)

Consider a fixed circle C with center O, called the fundamental circle. The hyperbolic points of H are the points interior to or inside circle C. The hyperbolic lines of H are the circular arcs orthogonal (or perpendicular) to C and the diameters of C. In the figure above, P, Q and O are hyperbolic points.

Why are R, S, A and B not hyperbolic points? Arc RS and diameter AB are infinite hyperbolic lines since points A,B,R and S are excluded, whereas, OP is a line segment. In Figure b, hyperbolic lines RS and T intersect at point P and both are parallel to hyperbolic line a. This illustrates the P5: “Through point P not on line a, more than one line (RS and TW) can be drawn parallel to line a” Non-intersecting lines are said to be parallel. In Figure c, a triangle LMN, with sides LM, LN, MN and angles at points L, M, and N. An interior angle of the triangle is measured by the magnitude of the angle formed by the tangents to the two side of the common vertex, shown in Figure d.

All Euclid’s axioms for plane geometry are satisfied in H except the axioms of parallels. Theorems that hold for both Euclidian and hyperbolic geometries are; those congruence of triangles; if P is a point not on line 1, only one line can be drawn through P perpendicular to 1; vertical angles are equal, and an exterior angle of a triangle is bigger than either remote interior angles.

Theorems in hyperbolic geometry that differ significantly from than of Euclid’s:1. The sum of the angles of a triangle is always less than 180o

2. Similar triangles are congruent.3. The area of a triangle is determined by the sum of its angles; particularly the bigger

the triangle is the closer the sum of its angles is to zero.4. No quadrilateral is a rectangle.

A O

B

RP

a

L

NMP

Q

In hyperbolic geometries, we merely can assume that parallel lines carry only the restriction that they don’t intersect. Furthermore, the parallel lines don’t seem straight in the conventional sense. They can even approach each other in an asymptotically fashion.

Elliptic GeometryAs in Poincare’s model for hyperbolic geometry which utilized Euclidian circles on the plane, we shall “construct” the model for plane elliptic geometry on the basis of Euclidian solid geometry. This refers to the surface of the earth which will be used to illustrate and interpret the Riemmanian axioms and theorems.

An elliptic line is a great circle. A great circle on a sphere is any circle whose diameter is equal to the diameter of the sphere.

Riemann’s version of Postulate 2 is interpreted in terms of a great circle which is indefinite or is bounded. One can travel around it without having to stop to a particular point, although it is of finite length (approximately 24,000 miles for the earth). It means further that in a given model all elliptical lines have the same length.

Now on the subject of parallelism, the fifth postulate of Euclid is replaced by the statement that the two elliptical or Riemmanian lines are parallel.

Points P and Q in the figure above are known as antipodal points. Antipodal points are diametrically opposite on the surface of the earth are regarded as “identical” points in order that Euclid’s postulate stating that “two points determine a straight line” will hold. Otherwise, two points like north and south poles lie on infinitely many great circles.

With the changes in Euclid’s postulates 2 and 5, theorems that differ from their Euclidian counterparts have evolved. Below are some of these theorems.

1.1 All lines perpendicular to a given line meet at a point NP

Equator

Let the equator be a line. The great circles perpendicular to the equator are the lines of longitude. Since all lines of longitude meet at the North Pole, all perpendiculars to the equator meet at a point.

b p

a

1.2 The sum of the angles of a triangle is always greater than 180o. From the adjoining figure, let points A,B and C be locations of Christmas Island, Ghana and the North Pole, respectively. A triangle is thus formed by points A, B, and C. From the world map, A and B lie on the equator and the great circle passing through the North Pole and Ghana is the 0o line of longitude (Ghana is aligned with Greenwich, England) while Christmas Island is on the 160o line of longitude. Computing the angles at A, B and C gives 90o, 90o and 160o respectively. The sum of the three angles of the triangle is 340o, which is more than 180o.

C

A B equator

Since any two points on the earth lie on a great circle, a triangle would have arcs of great circle as sides. If the triangles are very small relative to the earth’s surface, like the survey of a triangular farm, the sides of the triangle would look like straight lines and the sum of the angles of the triangle is almost 180o. As a matter of fact, the area of the small triangle, relative to the surface of the earth is almost zero. As it increases in size, their sizes are curves (or arcs) and the sum of its angles becomes increasingly larger than 180o.

1.3 Any two triangles with the same angle sum have the same area and the area of a triangle is completely determined by the sum of the angles of a triangle.

1.4 A true test for congruent triangles is AAA (i.e corresponding angles are equal)Elliptical geometry is useful especially to pilots and ship captains as they navigate around the spherical Earth. They able to know the shortest travel done. For example, the shortest flying distance from Florida to the Philippine Islands is a path across Alaska. The Philippines are South of Florida so it is not apparent why flying North to Alaska would be shorter. The answer is that Florida, Alaska, and the Philippines are collinear locations in elliptical geometry.

Projective GeometryProjective geometry deals with the formalization of one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way. A projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines will meet on a horizon in virtue of their possessing the same direction.

Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity.

Properties:Points are mapped to points and lines to lines.Parallel lines in three spaces which are not parallel to the drawing plane must be drawn to converge at their vanishing points.Desargues’ Theorem of Homologous Triangles

Theorem. Suppose there is a point O and triangles ABC and A'B'C' in the plane or three space. If they are projectively related from the point O, that is, the triples {O, A, A'}, {O, B, B'} and {O, C, C'} are all collinear. Then the points of intersections of the corresponding sides AB and A'B', AC and A'C' and BC and B'C' (or their prolongations) are collinear. Conversely, if the three pairs of corresponding sides meet in three points which lie on one straight line, then the lines joining corresponding vertices meet at one point (are projectively related.)

Points, rays, lines, and planes

A line can be represented in space by the equation ax + by + c = 0. If we treat a, b and c as the column vector and x, y,1 as the column vector then the equation above can be written in matrix form as:

or Or using vector notation

or

sweeps out a plane that goes through zero in and sweeps out a ray ( a ray goes through zero).

The plane and ray are subspaces in . A subspace always goes through zero.Ideal points

Applications of non-Euclidean Geometries:Non-Euclidean geometries began to replace the use of Euclidean geometries in many contexts. For example, physics is largely founded upon the constructs of Euclidean geometry but was turned upside-down with Einstein's non-Euclidean "Theory of Relativity" (1915). Newtonian physics, based upon Euclidean geometry, failed to consider the curvature of space, and that this constituted for major errors in the equations of planetary motion and gravity.

Also, a non-Euclidian Geometry especially elliptic geometry is applied to navigation and aviation, specifically when dealing with astronomical distances. It provides the description of the body of knowledge for the large scale spatial structure of the actual universe

REFERENCES:

CASTILLO et al. 1998. Notes in Mathematics for General Education. UP Diliman GE Mathematics Committee. pp. 15-20.

Circumference of a circle. http://en.wikipedia.org/wiki/Circumference. Date Accessed: July 21, 2007.

GLOSSER, G. 1998-2007. Math symbols created with MathType. Design Science, Inc.

Microsoft ® Encarta ®. 2006. Trigonometry. 1993-2005 Microsoft Corporation©.

Non-Eucledian Geometry. http://www.geocities.com/capecanaveral/7997/noneuclid.html. Date Accessed: 2007 July 21.

Sin, cos and tan functions. http://www.syvum.com/math/trigo/trigonometry1.html#T2. Date Accessed 2007 July 27.

Perimeter. http://en.wikipedia.org/wiki/Perimeter. Accessed July 21, 2007

Projective Geometry. http://en.wikipedia.org/wiki/Projective_geometry. Date Accessed: 2007 July 21.

Similarity. http://en.wikipedia.org/wiki/Similarity_(geometry). Date Accessed: 2007 July 22.

Similarity. http:/library.thinkquest.org/20991/geo/spoly.html. Date Accessed: 2007 July 22.

Trigonometry and right triangles. <http:// id.mind.net/~zona/mmts/trigonometryRealms/introduction/rightTriangle/trigRightTriangle.html> Date accessed 2007 July 27.