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Lines, Parallel and Perpendicular Lines, Parallel and Perpendicular

In mathematics, the term "straight line" is one of the few terms that is left undefined. However, most people are comfortable with this undefined concept, which can be modeled by a pencil, a stiff wire, the edge of a ruler, or even an uncooked piece of spaghetti. Mathematicians sometimes think of a line as a point moving forever through space. Lines can be curved or straight, but in this entry, only straight lines are considered. A line, in the language of mathematics, has only one dimensionâ€”lengthâ€”and has no end. It stretches on forever in both directions, so that its length cannot be measured. When a line is modeled with a piece of spaghetti, a line segment is actually being represented. The model of a line segment has thickness (or width), while the idea that it modelsâ€”a mathematical lineâ€”does not. So a mathematical line is a notion in one's mind, rather than a real object one can touch and feel, just as the notion of "two" is an idea in one's mindâ€”that quality and meaning that is shared by two apples, two trucks, and the symbols //, 2, â˜ºâ˜º, and ii. Think of two straight lines in a plane (another undefined term in geometry ). Someone can model this idea, imperfectly, by two pencils or two pieces of spaghetti lying on a desktop. Now, mentally or on a desktop, push these lines around, still keeping them on the plane, and see the different ways two lines can be arranged. If these two lines meet or cross, they have one point in common. In the language of mathematics, the two lines intersect at one point, their point of intersection. If two lines are moved so that they coincide, or become one line, then they have all of their points in common. What other arrangements are possible for two lines in a plane? One can place them so that they do not coincide (that is, one can see that they are two separate lines), and yet they do not cross, and will never cross, no matter how far they are extended. Two lines in the same plane, which have no point in common and will never meet, are called parallel lines. If one draws a grid, or coordinate system, on the plane, she can see that two parallel lines have the same slope, or steepness. Are there any parallel lines in nature, or in the human-made world? There are many models of parallel lines in the world we build: railroad tracks, the opposite sides of a picture frame, the lines at the corners of a room, fence posts. In nature, parallel lines are not quite so common, and the models are only approximate: tracks of an animal in the snow, tree trunks in a forest, rays of sunlight. The only other possible arrangement for two lines in the plane is also modeled by a picture frame, or a piece of poster board. Two sides of a rectangle that are not parallel are perpendicular . Perpendicular lines meet, or intersect, at right angles, that is, the four angles formed are all equal. The first pair of lines in part (a) of the figure below meet to form four equal angles; they are perpendicular. The second pair in part (b) forms two larger angles and two smaller ones; they are not perpendicular. Perpendicular lines occur everywhere in buildings and in other constructions. Like parallel lines, they are less common in nature. On a coordinate system, two perpendicular lines (unless one of them is horizontal) have slopes that multiply to a product of -1; for example, if a line has a slope of 3, any line perpendicular to it will have a slope of -â…“. see also Lines, Skew; Slope. Lucia McKay Anderson, Raymond W. Romping Through Mathematics. New York: Alfred A. Knopf, 1961. Juster, Norton. The Dot and the Line: A Romance in Lower Mathematics. New York: Random House, 1963. Konkle, Gail S. Shapes and Perception: An Intuitive Approach to Geometry. Boston: Prindle, Weber and Schmidt, Inc., 1974.

From Yahoo Answers

Question:Write a proof of the fact that the shortest line segment joining a point to a line is the perpendicular line segment. \ Hint: Start with a line L and a point P. Suppose you have the perpendicular segment from P to L, and you have any other segment from P to L (as in the diagram). Your goal is to show that the perpendicular one is shorter. You can use any of the results we proved or mentioned in class so far.)

Answers:Proof by contradiction (to follow it, draw pictures). Let A be the point on L such that the segment PA is perpendicular to L. Suppose there is another point, A', on L such that PA' is shorter than PA. (We shall see that this supposition leads to a contradiction, and this will complete the proof.) By the above construction, PAA' is a right triangle, with the right angle at vertex A. In this right triangle, therefore, PA' is the hypotenuse, and PA is a leg. By Pythagorean Theorem, the leg cannot be longer than the hypotenuse. Thus, PA cannot be longer than PA', a contradiction. This completes the proof.

Question:

Answers:If a line is perpendicular to another, it goes in the exact opposite direction, therefore creating a 90 degree angle at the intersection. The slope of the line(VW) is the negative reciprocal of the slope of the other(TU). ex: the negative reciprocal of 2 is -1/2 because if you make 2 into a fraction (2/1) then switch each number (to 1/2) it is the reciprocal, then, if it is negative(originally) it becomes a positive, and if it is a positive (originally) it becomes a negative.

Question:I need to write and equation of the perpendicular bisector of segment AB. A(0,0) B(6,0) I also need help in finding two points on the perpendicular bisector of C(0,0) D(0,4) Thank you

Answers:The line segment joining AB is on the y axis(vertical line) with midpoint (3,0). Since it must be perpendicular that makes the bisector horizontal through (3,0). It's equation is y = 3 the line segment joining CD is on the x axis (horizontal line) with midpoint (0,2). Since it must be perpendicular that makes the bisector vertical through (0,2). It's equation is x = 2. Any points on this line must have 2 as its x-coordinate such as (2,5) (2, -3)

Question:Also need to know how to make an angle bisector using only a right triangle tool. I've only done constructions with a compass and my teacher didn't teach me this so I'm confused...I'm not like a slacker I just really don't get it. Any help would be appreciated. :) (Didn't know if I should categorize this in homework or math so I put homework. If anyone's mad I'll delete it)

Answers:You could align the segment horizontally, lay one leg of the right triangle tool above and along the segment so that the right angle vertex is at one endpoint then draw the other leg straight up, then flip it and do the same leg straight down at the other end, then connect the two ends; this will bisect the segment, then make the perpendicular line through that midpoint with the tool. And the angle bisector, put one of the acute angles of the tool at the vertex of the angle to be bisected, line up the leg with one side of the angle, then draw in the other leg of the right triangle (which will be perpendicular to the side of the angle to be bisected; then flip the tool and do the same to the other side of the angle to be bisected; where the two lines you drew cross, that's there the bisector will go through. This would have been way easier to draw rather than type- good luck figuring it out!

From Youtube

Drawing a Perpendicular Bisector :A brief video made on a SMART Board describing the steps to drawing a perpendicular bisector which divides a line segment into 2 congruent parts.

To Construct the Perpendicular Bisector of a Line Segment :We used a compass and ruler to bisect a line segment.