Similarity of Right Triangles
Two right triangles are similar if they have an angle in common BESIDES a right angle. More generally, two triangles are similar if and only the angles in one equal the angles of the other in some order.
For similar triangles, the ratio of their matching or corresponding sides are equal. The diagram below illustrates this in a special case.
The observation or assumption that
the ratios of side lengths, here adjacent/hypotenuse or opposite/hypotenuse are independent of the scale factor and depend only the angle the hypotenuse makes with the adjacent side
justifies the definition and calculation of trig functions using the ratio of sides for similar right triangles.
Slopes for Straight Lines
The assumption that the ratio m = rise/run is a constant for a straight line provides the basis for what was the discussion of slopes in algebra or pre-calculus courses. For each straight line, its slope m is the proportionality constant linking the rises to runs for any two points on the line.
The rest of this webpage is a digression
Linear Algebra and Slopes of straight lines.
If solving simultaneous equations is easy for you, see the chapter on this in the section Three Skills for Algebra, then a large part of high school equatiion solving becomes very simple -- too simple, as then solutions to problems become the task, hard or not, of extracting linear to solve from the information in a problem.
For any two points [p,q] and [u,v] on a straight line with slope m, the rise = v - q and the run = u - p. The assumption m = rise/run gives the equation
m = (v-q)/(u-p)
Here is formula for calculating m given two points on the line. Text books prefer to use x's and y's with subscript to denote the two points. I leave the change of notation to you.
Next multiplication of both sides by (u-p) gives m(u-p) = v -q or v = m(u-p) + q.
Suppose the slope m of the line and a point (p,q) on the line are given. Further suppose [u,v] =[x,y] is unknown. Then we assume the point with coordinates [x,y] is on the line if and only if
y = m(x -p) + q
This represents the point slope form of the equation of the line. This if and only if assumption is needed to link coordinates with geometry. Write when and only when instead, if you like :)
If (p,q) happens to be on the x-axis (the horizontal axis) then q = 0 and the foregoing equation becomes
y = m(x -p)
The previous equation might be called the x-intercept equation for a line. For coordinates [x,y] satisfying an equation of this form, taking x to p forces y = 0.
If (p,q) happens to be on the y-axis (the vertical axis) then p = 0, the equation becomes
y = mx + q
The latter is called the y-intercept form of the equation of a line. For coordinates [x,y] satisfying an equation of this form, taking x to be 0 forces y = q.
Algebraic Games with Equations for Straight Lines
When a point [x,y] is on two lines, it must satisfy the equations of both. This observation leads to two "linear" equations in two unknowns
ax + by = e
cx + dy = f
These simultaneous equations must have a unique solution if the lines are different and not parallel. The equations of lines y = something can be rewritten in the form ax+by = e with b = 1.
The solution of these equations can be complicated by requiring you the learner to obtain them from data such as two points on a line, or slope and point, or slope-intercept information. So before you solve two equations to find the intersection, if any, of two lines, you have to find a chain of reason which gives the coefficients of the equations. Exercises with this are met or avoided in the high school or college "theoretical" development of chemistry and physics, and other subjects too.
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Complex Numbers |
The fundamental theorem of algebra and partial fraction decomposition in calculus depend on complex numbers.
Easy Consequences
Hint: See the (newest) Complex
Number. Starter Lesson.
for a simple geometric introduction, then continue with easy consequence below.
The clearest geometric proof of the distributive law appears in the
folder.
First Earlier (Old) exposition of complex numbers follows in Z1 to B1 below - read for review or revision .
D1 to D6 after provide a review of vectors.
More on Complex Numbers:
Chapters in Volume 3::
Intro assumes the field properties
of real numbers in place of
deriving them geometrically