similar triangles and trigonometry

8/13/2019 Similar Triangles and Trigonometry

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www.andrews.edu/~calkins/math/webtexts/geom13.htm#SIM

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A Review of Basic Geometry- Lesson 13

Similar Triangles and An Intro. to Trigonometry

Lesson Overview

Similar Triangles

Geometric Mean

Special Triangles, Side Length Ratios, and Trigonometry

Vectors, Vector Products, and Vector Spaces

Homework

Last chapterwe introduced the similarity transformation and this chapter we will apply it more specifically to

triangles. Similarity between triangles is the basis of trigonometry, which literally means triangle measure.Asnoted inNumbers lesson 11,the trigonometric functions can be thought of as ratios of the side lengths in right

triangles. Please review the informative paragraph and table of special trigonometric values given there.

Similar Triangles

If three sides of a triangle are proportional to the three sides of another triangle,

then the triangles are similar (SSS Similarity Theorem).

If the angles (two implies three) of two triangles are equal,

then the triangles are similar (AA Similarity Theorem).

Note: thisapplies not only to ASA, AAS=SAA, but also to AAA situations.

If two sides of a triangle are proportional to two sides of another triangle and the

included angles are congruent, then the triangles are similar (SAS Similarity Theorem).

Thus remains the SSA (ASS) case, which remains ambiguousunless HL or SsA occurs.

Lines parallel to a side of a triangle intersect the other two sides at nonvertices,

if and only if the two sides are split into proportional segments.

Since this theorem is given as an if and only if, it goes both ways. Thus the textbook has both a theorem and its

converse.

Geometric Mean

We introduced the geometric mean somewhat in the last chapterand somewhat in statistics.Please review what

we have there. The geometric mean is typically first encountered in aproportionwhen the means are equal, as in
http://www.andrews.edu/~calkins/math/webtexts/geom12.htm#GMhttp://www.andrews.edu/~calkins/math/webtexts/stat04.htm#GEOhttp://www.andrews.edu/~calkins/math/webtexts/numb11.htm#TRIGhttp://www.andrews.edu/~calkins/math/webtexts/geom12.htm#PROhttp://www.andrews.edu/~calkins/math/webtexts/stat04.htm#GEOhttp://www.andrews.edu/~calkins/math/webtexts/geom12.htm#GMhttp://www.andrews.edu/~calkins/math/webtexts/geom07.htm#SsAhttp://www.andrews.edu/~calkins/math/webtexts/numb11.htm#TRIGhttp://www.andrews.edu/~calkins/math/webtexts/geom12.htmhttp://www.andrews.edu/~calkins/math/webtexts/geom13hw.htmhttp://www.andrews.edu/~calkins/math/webtexts/geomtoc.htm


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8/w=w/4. Here w2=32 and square rooting both sides gives an answer. However, in general, there may be nnth

geometric means. We thus cannot be sure of the sign of wabove.

The geometric mean is developed here because of its application to right triangles and the way the altitude to the

hypotenuse divides the triangle into similar triangles. Assume you have two of the three terms in a geometric

sequence, such as 2, ?, 50. In other words, you want some numberg, such that 2/g=g/50, or 250=100=g2, so

obviously,g=10 or perhaps g=-10. Often the positive geometric mean is required and will be so specified.

The use of lower case letters a, b, and cfor the sides of a

triangle is a common convention dating back to Euler

which we will adhere to. arefers either to the set of points

composing the side or the length of the side, depending on

context. The angle opposite side ais A, the angle opposite

side bis B, and the angle opposite side cis C. If it is a

right triangle, Cwill be right so cwill be the [length of the]

hypotenuse. Given the right triangle ABC with height h

(CD) to the hypotenuse, h= (xy), whereas a= (cx), and b= (cy). Here x+y=c(BD + AD = AB), and yis

the leg of a similar triangle with hypotenuse b, and xis the leg of a similar triangle with hypotenuse a.

The altitude of a triangle is the geometric mean of the segments of the hypotenuse that it divides.

Each leg would also be a geometric mean of the hypotenuse and the adjacent segment of the hypotenuse.

Special Triangles, Side Length Ratios, and Trigonometry

An isosceles right triangle (454590) is a very special triangle. Its side lengths form a very special ratio which

must be memorized. Specifically, if the legs are both of lengthx, then the hypotenuse is of length xby the

pythagorean theorem. This ratio of 1/ = /2 or about 0.707 must become familiar.

Similarly, the 306090triangle must be memorized, somehow. One way is to start with an equilateral

triangle, bisect one angle which also bisects the side opposite, and consider the resulting congruent triangles.

Obviously, two congruent 306090triangles are formed. Again, by the pythagorean theorem, the side length

ratios can be found to be 1: :2. By the AA Similarity Theorem, any triangle with these angles has these

exact side length ratios.

In right triangles, the six side length ratios are proportional to the angles and named as follows.

sinA= opposite leg hypotenuse csc A= hypotenuse opposite leg

cos A= adjacent leg hypotenuse sec A= hypotenuse adjacent leg

tanA= opposite leg adjacent leg cot A= adjacent leg opposite leg

Various mnemonics are commonly employed to assist in the recall of these ratios. SOH-CAH-TOA, with some

apocryphal reference to a so named Indian chief is common: S=Sine, O=Opposite, H=Hypotenuse, C=Cosine,

A=Adjacent, T=Tangent. Oh Hech Another Hour Of Algebra is another similar mnemonic. Another one only

recently brought to my attention is: OHAHOAAO, which is short for Oscar Had A Hand On Alice's Arch Once

which gives, in order, the ratios for Sine, Cosine, Tangent, and Cotangent. Note how the right hand column


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reciprocates the left hand column. For this reason, the left hand column are considered the primary trigonometric

functions and the right hand column secondary. Most calculators only have the primary trig functions.

Consider a right triangle with c=1, one vertex at the origin, one side along the x-axis, and the right angle formed

by a perpendicular to the x-axis, thus the hypotenuse lies in quadrant I. In this case we often label one leg yand

the other legx. Consider what happens as we let the angle at the origin change. As the angle increases, the side

opposite increases. Since we fixed c=1, that vertex traces out a portion of a unit circle. The sine function

describes how this side changes with angle, starting at 0 for 0

and increasing monotonically to 1 for 90

(ofcourse at 0and 90it really isn't a triangle). Similarly, the cosine function, which describes how the adjacent side

varies, starts out at 1 for 0and is always decreasing to 0 for 90.

Tangent is not only the ratio of the opposite side to the adjacent side, but can also be written as sine over cosine.

If cosine equals 0, we have a problem, division by zero, so the tangent of 90is undefined. Large tables of trig

functions were commonplace before calculators became ubiquitous. Below is a small table in 5increments

between 0and 90. Notice how the cosine values are the same as the sine values of the complementary angle.

Tangent and cotangent are similarly related. A later lesson will extend these trig functions past a right angle, into

negative angles, graph them, and reintroduce the unit circle.

Angle

(degrees)

Angle

(radians) Sine Cosine Tangent Cotangent Secant Cosecant

0 0 .0000 1.0000 .0000 NAN 1.0000 Infinite

5 /36 .0872 .9962 .0875 11.4301 1.0038 11.4737

10 /18 .1736 .9848 .1763 5.6713 1.0154 5.7588

15 /12 .2588 .9659 .2679 3.7321 1.0353 3.8637

20 /9 .3420 .9397 .3640 2.7475 1.0642 2.9238

25 5 /36 .4226 .9063 .4663 2.1445 1.1034 2.3662

30 /6 .5000 .8660 .5774 1.7321 1.1547 2.0000

35 7 /36 .5736 .8192 .7002 1.4281 1.2208 1.7434

40 2 /9 .6428 .7660 .8391 1.1918 1.3054 1.5557

45 /4 .7071 .7071 1.0000 1.0000 1.4142 1.4142

50 5 /18 .7660 .6428 1.1918 .8391 1.5557 1.3054

55 11 /36 .8192 .5736 1.4281 .7002 1.7434 1.2208

60 /3 .8660 .5000 1.7321 .5774 2.0000 1.154765 13 /36 .9063 .4226 2.1445 .4663 2.3662 1.1034

70 7 /18 .9397 .3420 2.7475 .3640 2.9238 1.0642

75 5 /12 .9659 .2588 3.7321 .2679 3.8637 1.0353

80 4 /9 .9848 .1736 5.6713 .1763 5.7588 1.0154

85 17 /36 .9962 .0872 11.4301 .0875 11.4737 1.0038

90 /2 1.0000 .0000 NAN .0000 Infinite 1.0000


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Note especially NAN for tan 90=cot 0. NAN is FORTRANese for Not A Number. It may look like it

approaches positive infinity, but since it will approach negative when approaching 90from above, and since the

two limits do not agree, we say it is not a number or undefined.

Example:Consider a flagpole, set precisely perpendicular to the ground, which casts a shadow 11.3355m long

and the angle from the ground up at the tip of the shadow to the tip of the pole is 34.5543 . What is the height of

the flagpole to correct significance? Solution:Tangent is the ratio of the side opposite the given angle to the side

adjacent, thus tan 34.5543=h/11.3355m. Solving for hwe obtain h=11.3355mtan 34.5543=7.80650m. Ifyou obtained a negative answer, change your calculator to degrees mode!

The area of any triangle can be found by splitting it into two right triangles.

Area of a triangle = (1/2)absin C, where Cis the angle between sides aand b.

Vectors, Vector Products, Vector Spaces

Math and physics often classify objects by dimensionality, with scalars(magnitude only) and vectors

(magnitude plus direction) occupying the bottom rungs. A classic physics example involves speed (a scalar) and

velocity (a vector)speed tells how fast an object is going, but not in what direction.

One way to develop vectors uses the cartesian coordinate system and ordered pairs or triples. A vector is then

defined by the directed line segment from the origin to the given point. The end at the origin is the tail,whereas

the other end is the headand often an arrowhead it put there. This gives a certain direction between [0 ,360)

and a certain magnitude or length which can be calculated via the distance formula. The vector represented by

the directed line segment between (0,0) and (3,4) is the same as the one between (1,1) and (4,5), however,

since these have the same magnitude and direction. We often say we can move the vector around as long as we

don't change the direction it points or its length. We often break vectors down into x, y, andzcomponents,especially using sines and cosines for the yandxcomponents. Unit vectors, vectors of length 1, especially in

thex, y, and zdirections are also utilized. There are special names for these: i=(1,0,0),j=(0,1,0), and k=(0,0,1)

A unit vector in another direction, such as v=(1,1,1) can be obtained by dividing this vector by its length

designated by |v|: (1,1,1)/ , where it is understood that the scalar applies to all components of the ordered

triple.

Vectors are said to be added head to tail, or if given in component form, via the components. Thus the sum of

the vectors (x1, y1, z1) and (x2, y2, z2) is the vector: (x1+x2, y1+y2, z1+z2). Pictorally, the sum is the diagonal of

the parallelogram formed by adding them in either order. [make diagram]

The direction of a 2-D vector can be obtained from its xandycomponents. Specifically, the tangent of the angle

relative to thex-axis is equal to y/x. Your calculator has a special key tan-1, better known as atan, to find this

arctangent or inverse function. DO NOTconfuse this with reciprocal. (It must become clear to you by context

whether arctangent or cotangent is meant.) Some adjustment of the calculated value will become necessary since

results for quadrants I and IV are assumed. Since our textbook restricts angles to between 0and 180, we will

defer this into another lesson.

The scalaror dot productof the vectors (x1, y1, z1) and (x2, y2, z2) is the scalar quantity:x1x2+ y1y2+ z1z2.


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The vectoror cross productof two vectors is a more complicated entity which we will leave for a later lesson.

Modern physics and mathematics is firmly based on the algebra of vector spaces.

BACK HOMEWORK ACTIVITY CONTINUE

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URL: http://www.andrews.edu/~calkins/math/webtexts/geom13.htm

Copyright 2004-05, Keith G. Calkins. Revised on or after May 8, 2005.
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