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Pependicles
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If you want to know more about the perpendicular, you're in the RIGHT place, read on and find out you knew it all along.
What's right is right.
Along the right line
Plumb Bobs and uprights
This is where it's at! This is both the proof for a straight line and the fundamental understanding of the 2 reference frames in which we all live, vertical and horizontal, these 2 right lines and their relationship to one another is what,(in my opinion) is the building block upon which all practical geometry is based.
Perpendicularity
Right lines make right angles
Before there was straight there was right, two right lines have a relationship with each other.
A right line is a straight line and as such can form a right angle to another straight line making all right angles equal to one another, Axiom 11. Only a straight line has the ability to do this and the right angle produced by this relationship is the basis of how we reference the vertical and horizontal, these are the 2 reality frames of reference that we all know and use on a daily basis, from the balance we feel through the ears or the entire understanding of the horizon being horizontal, these 2 facts are born out in both Euclidean geometry and reality, geometry is the measure of the earth after all, the link is undeniable.
Sine, cosine, and tangent are functions of angles that are defined as different ratios of the sides and hypotenuse of a right triangle that include that angle. The relationships between them take many forms based on their definitions and our knowledge of other truths about triangles, including, most importantly, the Pythagorean Theorem.
Pythagoras' theorem states that, in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides.
If the sides of the right-angled triangle are labelled a, b and c then Pythagoras' theorem states: c2=a2+b2
If a is a radian angle (in R), then the other people's proofs suffice. It must be said, however, that historically, tan a was not defined as opposite / adjacent, but the length of the tangent from the point on the circumference of a unit circle meeting the angle a in the centre of the circle to the positive horizontal... hence the name 'tangent'!
