Pythagoras theorem

 

You have your mathematical instinct. All you will need is your operating system's paint or image editor program, calculator and a geometry set.

In the line drawing mode select the right angle triangle from the shapes menu. Using your mouse, ( or the touch screen )  the software will allow you to drag a copy into the canvas. The program allows to adjust any size you like. Holding down the mouse button and dragging round the canvas or the touch  screen illustrates the various options of the triangle can be, from tall, wide, top, bottom, left or right pointing options.

Set the vertical and horizontal lines equal lengths.

Using the insert text click opaque and type in the center of the angled line the letter h for hypotenuse.

 Do the same with  the vertical side, typing o for the opposite side and in the horizontal line a for the adjacent side.

With a ruler measure each line using insert text type a list under the illustration o equals ( o = ) a equals ( a = ) and h equals the measurements you recorded each line. This is for safekeeping.

Note multiplying numbers by themselves are called squares. The shorthand arithmetic notation is a little razed 2 just to the right of the number. The next step involves a squaring processes on each measurement you recorded. Unless you have a genus memory we need to write results down for safe keeping.

Using insert text type in a reserved space o squared equals the recorded measurement , a squared equals the recorded measurement and h squared equals the recorded measurement respectively. Make another list, but this one a squared equals, o squared equals and h squared equals respectively.

Now add them typing in the total addition equals, recording down for safe keeping.

The law of mathematics tells us if we divide the total by itself we will get the root of the square called the square root. Calculators have a tick like key in the key pads. Enter the total of the addition pressing the square root key.

When you come to measure the h line with a ruler it will agree with the calculator.

In mathematical syntax expression h equals the square root of a squared plus o squared.

Turning your attention to the geometry kit use a full circle or better if a half circle protractor lining up the vertical line (  the 90 degree centre of the protractor line ) with the vertical  line of the triangle and the horizontal line of the protractor with the horizontal line of the triangle. Mathematics tells us horizontal lines are not angles. The protractor agrees a is 0 degrees , o a 90 degree angle and measures the angle of h.

If the o and a lines are equal the protractor will agree h equals half of  90 degrees ( a 45  degree angle ). Because of the effects of changing the triangle effects the text  so fresh diagrams may be required. Operating on the principle you can apply the latter to many different wide or tall triangles. Applying the Pythagoras theorem agrees it can be applied to a infinite number of right angle triangle options.,

The theorem allows us to to use the square root calculate the length of the h line. To put into perspective using numbers suppose 3 ( representing units in centimeters or inches if you like ) for a and 4 for o. Your mathematical instinct agrees a is three 3's is 9 and o is four 4's is 16.

We have a equals 9 and o equals 16. Then 16 plus 9 equals 25. The square root of 25 is 5. This is the famous 3-4-5, c squared equals a squared plus b squared formulae. Thus h ( c or h if you like ) equals 5. The square root of 25 projects c as 5. In other words c squared equals the square root of a squared plus b squared. If you draw up a triangle 3 centimeters ( or in inches if you like ) for a and 4 for o checking with your ruler ruler will agree c equals o plus 1. The law of mathematics tells us h minus 1 equals o.

If we use double 3 to 6 for a and double 4 to 8 for b 6 squared plus 8 squared equals 100. Of course when we square root it equals 10. When you measure a 6 centimeter high by 8 centimeter long right angle h line will agree. Your mathematical instinct spots an interesting paten the triangle is exactly a scaled double the size of the 3-4-5 triangle.

Other examples include a 5 centimeter tall and 12 centimeter long right angle expressed as 5-12-13. Five 5's is 25. Twelve 12's is a144. A 144 plus 25 is a 169. The square root of 169 is 13. Our mathematical instinct spots an interesting coincidence where h happens to equals 12 plus 1 and 12 equals h minus 1 respectively.

You will find there are a infinite number of examples.

We can apply the theorem  to a flag pole. Fine days cast's a shadow. The protractor aggress with mathematics telling us the vertical up right angle of the pole is a 90 degree angle while math's tells us the shadow as in all horizontal lines is not an angle, thus a 0 degree, and an imaginary line at the tip of the shadow straight up to the top of the pole the hypotenuse line.

The same triangle theme of a board leaning against a wall in this case the wall a, the floor at right angles to it o and the board represents h.

The flag pull is the tall of a right angle and the shadow the length. The pole's shadows is the shortest in the early morning, the longest midday and late afternoon shrunk back to the shortest an example of the infinite variations to the right angle. Thus as the shadow varies the length at different times of the day thus varying the angle of h. If a pole is telescoped it to effects o and h. This can be equally applied to a tall buildings and engineering right angle braces such as supporting beams of bridges and other structures. Right angle triangle braces are commonly seen in right angle truck chassis beams.

When the shadow of flag poles equals the total length of the pole the h angle is half of 90 degrees. If the shadow is shorter changes the angle. can check adjusting a simple triangle uneven 0 and a measuring h with a protractor.

Your mathematical instinct easily spots a pattern protractors confirming vertical lines are always 90 degrees and the horizontal lines not angles ( 0 degrees ) both left and right of the vertical line reference. Your instinct will recognizes two 90's equals a 180 degrees a half circle. It will recognize two more 90 degree right angles that pans out to four 90 degrees equals a full 360 degrees circle. It easily spots a pattern of  90 degrees is a 1/4 circle, plus 90 equals 180 a 1/2 circle plus another 90 degrees 270 degrees 3/4 circle, plus another 90 degrees a full 360 degree circle.

To proves this draw in a circle

Now draw a horizon through the center. This is the a line dividing the circle clean in half horizontally.

Draw in a vehicle line right down the center. This is 0 dividing the circle in half vertically

Your mathematical instinct easily spots a symmetry paten in two dimensional form o and a forms a cross in the circle forming equal upper left and right and lower right and left pointing corner right angle triangles. When h lines are dawn in concentrating where o and a lines intersect in the center of the circle your instinct should spot straight away an upper left and right and lower right and left pointing total of 4 ( equal later rule ) Equilateral triangles.

When you draw a vertical line thought the center of an equilateral triangle dividing it in half you will observe back to back right angle triangle.

The question is which are they right angles or equilateral? Your mathematical instinct will recognize can be both which ever point of view you work from. The illustration shows proof equals a right triangle when you test with the Pythagoras theorem formula can be applied to equilateral triangles.

If you divide o by h and take a protractor reading of h aggress called a sine angle. If you divide a by o by h is a cosine and if you divide o by a is a tangent angle. We have three angle equations, sine equals o over h, ( sine= a/h ), cosine equals o over h ( cos = o/h ) and tangent equals o over a ( tan = o/a ) respectively. You will note in scientific calculators sin, cos, and cosine keys.

Operating on the principle we can apply the law of mathematics we can apply arithmetic instead of checking with a protractor the formulas h times a equals the tan angle of the h line. h times o equals a sine of  and a times o equals the cosine of h.

The angles applies to navigation called a vector. In this case the right angle o line is a normally referenced as the north and south line ( n and s ) with arrow heads in each line showing direction of travel. The a line is automatically set east and west direction. ( e and w ). The angle arithmetic needs two known reference values. Any two the other can be found. Reference numbers for any of the the lines the third unknown can be projected. The letters a, b, c and x, y, and z is very common in in text books.

The h line expressed air current off setting airplane and water current on boats tending to drift off course. For example a swimmer crossing diagonally across a river is curried down street by a current ending up on the opposite bank further down stream. The h angle tells us how much by. Similarly wind drifting aircraft.

Distance units ( kilometers or miles if you like ) an a the length of time to cover the length of o ( hours units for example ) are number substitutes for o and a. Knowing o and a the angle h angle can be projected though the angles formulae. Vice versa knowing h and a, o can be projected just as knowing o and h. Algebra is a branch of a branch of mathematics that teaches us how manipulate the laws of mathematics using letters and symbols with out numbers applies to the right angle triangle calculations.

When you manipulate the triangle in your image editor you can visually see how this is done adjusting the angle of h. Your computer is calculating is tracking and calculating your every movement as you go as we do with the angles arithmetic.

The right angle triangle can narrow down pin pointing on maps expressed as triangulation. Reference two reference points are always required. Letters that look like English a, b, c and, x, y, z ( actually Greek letters ) are markers or coordinates at the tip of each corner of the triangle. The triangle is squared in to a square where the tracking object is somewhere inside that square.

Your image editor program will demonstrate this called vector graphics.

Draw a square.

Now draw in h ( a line from the top left or right corner or the top left down to the opposite bottom corner. Your mathematical instinct will recognize back to back right angle triangles.

By adjusting the h line making the line longer of shorter you're computer calculates the other four edges in equal proportion. If you double the h line the whole square is double the size proportionately.

If you half it the whole square is reduced in size proportionally.

We can apply percentage arithmetic on the h line calculating the percentage smaller or larger by lengthening or shortening with a reference percent number for the h line will automatically set the whole square proportionally. Checking all 4 lines of the square with your ruler will agree.

As you enlarge the right and triangle becomes clear. You mathematical instinct recognizes you can use the principle with just a right and triangle applying all the angle and Pythagoras theorem formulae guided by the laws of Algebra.