It is important to note that the equation of a line in three dimensions is not unique. Choosing a different point and a multiple of the vector will yield a different equation.
Take a sheet of A4 paper. Fold it in half from top to bottom. Turn it round and you have a smaller sheet of paper of exactly the same shape as the original, but half the area, called A5. Do this on a large A3 sheet and you get a sheet of size A4.
The sides must be in the ratio of 1: By the two sheets being of the same shape, we mean that the ratio of the short-to-long side is the same.
If we want the smaller piece to have the same shape as the original one, then, if the longer side is length f and the short side length 1 in the original shape, the smaller one will have shorter side of length f-1 and longer side of length 1. So the ratio of the sides must be the same in each if they have the same shape: Thus if the sheets are to have the same shape, their sides must be in the ratio of 1 to Phi, or, the sides are approximately two successive Fibonacci numbers in length!
Here is a decagon - a sided regular polygon with all its angles equal and all its sides the same length - which has been divided into 10 triangles. Because of its symmetry, all the triangles have two sides that are the same length and so the two other angles in each triangle are also equal.
In each triangle, what is the size of the angle at the centre of the decagon? We now know enough to identify the triangle since we know one angle and that the two sides surrounding it are equal. Which triangle on this page is it? From what we have already found out about this triangle earlier, we can now say that The radius of a circle through the points of a decagon is Phi times as long as the side of the decagon.
This follows directly from Euclid's Elements Book 13, Proposition 9. Penrose tilings Recently, Prof Roger Penrose has come up with some tilings that exhibit five-fold symmetry yet which do not repeat themselves for which the technical term is aperiodic or quasiperiodic.
When they appear in nature in crystals, they are called quasicrystals.
They were thought to be impossible until fairly recently. There is a lot in common between Penrose's tilings and the Fibonacci numbers. The picture here is made up of two kinds of rhombus or rhombs, that is, 4-sided shapes with all sides of equal length. The two rhombs are made from gluing two of the flat pentagon triangles together along their long sides and the other from gluing two of the sharp pentagon triangles together along their short sides as shown in the diagams below.
Dissecting the Sharp and Flat Triangles: Here the sharp triangle is dissected into two smaller sharp triangles and one flat triangle, the flat triangle into one smaller flat and one sharp triangle.
At each stage all the triangles are dissected according to this pattern. Repeating gives rise to one version of a Penrose Tiling.
Note that the top "flat" diagram shows the sharp and flat triangles have the same height and that their bases are in the ratio Phi:You can use slope relationships to write an equation of a line parallel to a given line. What is the equation of the line through a given point and parallel to the x-axis?
Why? Find the equation of the line that contains the given segment. AB _ BC _ AD _ CD _ if point C is at (7, 3).
We write ABjjCD to denote AB is parallel to CD. We We will contruct the line DE that contains point B and is parallel to AC. The existence of this line is guaranteed by the parallel pastulate.
B A C D E. Statements Reasons 1. 4ABC is a triangle 1. Given 2. Construct DE so that DE contains B . Significant Energy E vents in Earth's and Life's History as of Energy Event. Timeframe.
Nuclear fusion begins in the Sun. c. billion years ago (“bya”) Provides the power for all of Earth's geophysical, geochemical, and ecological systems, with the . MATH PARALLEL, PERPENDICULAR, KSU VERTICAL AND HORIZONTAL LINES Deﬂnitions: † Parallel Lines: are two lines in the same plane that never intersect.
† Perpendicular Lines: are two lines that intersect to form a 90– angle. † Vertical Lines: always have the equation x = c, for some constant srmvision.com example, the equation of the vertical line through (a;b) is x = a. Free parallel line calculator - find the equation of a parallel line step-by-step.
Find the Equation of a Line Parallel or Perpendicular to Another Line – Practice Problems Page 2 of 4 Detailed Solutions 1. Find the equation of a line passing through the point (4, –7) parallel to the line 4x + 6y = 9.