I am trying to do things "by the book" and set up something that resembles batterboards. I have read to make the corners square to use the 3 4 5 method.
So I put a stake in the corner and measure out three feet and put a stake one way and then measure four feet the other and put a stake and if I did it right the hypotenuse, or the distance between the two stakes at the three and four feet marks is five feet and this insures what? That I have straight lines going both directions? And I also read another way was simply to measure diagonals of the alleged square to see if they are equal. Do you guys do any of this? Thanks.
That the angle between the 3 foot line and the 4 foot line is exactly
90°, or "square."
The tension on the string does that.
You seem to be confusing the word "square." In the context of your original question, "square" means a right angle; an angle measuring
90°. If the meeting of two boards, two pieces of steel, two lines of string, etc., is 90°, then that sub-structure is square.
In the context of measuring diagonals, a square is a special form of a geometric figure (rectangle) that will have angles equaling 90° if the diagonals (distance from opposing corners) are equal. This does assume that opposing sides are equal (if they aren't, then the figure isn't a rectangle; it's a trapezoid; a subset of a geometric figure called a quadrilateral).
Finally, although you didn't further muddy the waters by mentioning it, a square is a device that is precisely manufactered to gauge angles at 90°.
In article , snipped-for-privacy@removethispartpobox.com (LRod) wrote: [snip]
Close but no cigar.
If the diagonals of a four-sided figure are equal, the figure is a rectangle, but it is not necessarily a square. If the diagonals are equal, and *adjacent* sides are equal, the figure is both a rectangle and a square. In either case, the angles at the corners are right angles (the literal meaning of the word "rectangle").
If *all* opposing sides are unequal, it's a quadrilateral (literally, "four sides") and *not* a trapezoid. A trapezoid is a four-sided figure consisting of two parallel but unequal sides, and two equal but non-parallel sides, i.e. ___ /___\
Finally, opposing sides being equal defines a parallelogram, not a rectangle. A parallelogram is a rectangle if and only if its angles are right angles. And before anybody jumps on me to say that a parallelogram is defined by
*parallel*, "not" equal, opposing sides, let me point out that it's trvial to show that the two are equivalent: each implies the other.
-- Regards, Doug Miller (alphageek-at-milmac-dot-com)
:-( You're quite right, of course -- that's what I get for posting before coffee. I *meant* to say, if diagonals are equal *and* opposite sides are parallel...
I stand corrected.
"FALSE TO FACT" yourself -- the proof that these are equivalent is absolutely trivial. Given a quadrilateral with opposing sides equal, construct a diagonal. The two triangles formed are congruent; hence, the interior angles at the diagonal are congruent, and the sides are parallel. Alternatively, given a quadrilateral with opposing sides parallel, again construct a diagonal; again, the triangles formed are congruent, and therefore their corresponding sides are equal.
And the distinction is of absolutely *no* importance to any *practical* considerations whatsoever, unless you happen to inhabit a non-Euclidean universe.
-- Regards, Doug Miller (alphageek-at-milmac-dot-com)
Ah, but we do, whenever we use a level to determine if something is "Flat". Granted, it would have to be something rather big that you are making to make much of a difference...
Disproof by "real-world" counter-example: Take a globe. lines of lattitude are perpendicular to the lines of longitude, therefore, by definition, they _are_ parallel to each other. lines of longitude are perpendicular to the lines of latitude. therefore, by definition, they _are_ parallel to each other.
Consider a quadrilateral with points at "0 W, 0 N", "60 W, 0 N", "60 W ,60 N", and "0 W, 60 N" all four corners are 90 degrees, opposite sides _are_ parallel, yet the 'north' side of the square is only 1/2 the distance of the equatorial side.
And, when you construct your 'diagonal', the triangles formed are *not* congruent. Despite the fact that all three angles, and two adjacent sides _are_ the same measure.
In _plane_ geometry, opposite sides of a parallelogram being of like length
*is* an inescapable 'side effect' of the definition.
You sir, "don't know what you don't know".
Anyone who engages in long-distance navigation on, or above, the surface of the Earth has to deal with such matters on a day-to-day basis. Because it _is_ non-Euclidean. "Spherical geometry", as a matter of fact. Where many things 'everybody knows' are simply incorrect. e.g., the sum of the angles in a triangle is always _more_ than 180 degrees. And the shortest distance between two points is *not* a straight line (rather, it is a 'great circle' route).
Unfortunately, _you_ demonstrate a lack of understanding of spherical geometry.
If the of longitude are perpendicular to the equator, then they are BY DEFINITION perpendicular, as well, to any lines that are parallel to the equator. As you admit the other lines of latitude -are- parallel to the one representing the equator, the lines of longitude are _also_ perpendicular to those other parallels.
That 'parallel lines never meet' is *NOT* part of the definition. And is true *only* in "Euclidean plane geometry".
They are both perpendicular to a common line, thus complying with the actual mathematical DEFINITION.
In 'spherical geometry', the "ground rules" _are_ different. A lot of what 'everybody knows', from _plane_ geometry, simply "doesn't apply".
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