How does the scale factor in Algebra/geometry work?
968074
The scale factor is used to enlarge or shrink a given figure.\nFor example, if a triangle has sides 4,6,and 8, you can create a new triangle by multiplying each side by 2 and get a larger triangle with sides 8,12, and 16. If you wanted a smaller triangle, you could have multiplied each side by a scale factor of 1/2 getting a smaller triangle with sides of 2,3,and 4. In either case the triangles would all be similar (have the same shape, but not the same size.\n\nYou can do the same with points in a coordinate system. For example (3,7) can bmultiplid b y a scale factor of 3 making its new position (9,21).
340934
I think there's several ways to conceptualize this stuff.  First of all, a number raised to a power doesn't necessarily involve any extra dimensions at all.  3^3 = 9, and 9 is a number that can be graphed on a one dimensional line.  Consider, what dimension does 4^1.5 exist on, the 1.5th dimension?  That doesn't make much sense.  The answer is simply 4^1.5 = 8, and 8 can also be graphed on a one dimensional line.  Or how about 3^pi?  Do we get the pi'th dimension?  Mmm, a dimension of pie...  Er, what was I saying?\n\nAnyway if we want to have a real cube in three dimensional space, we need at least three points in 3D space to represent it (actually we can represent it with 6 pieces of info instead of the 9 x,y,z's that make up the 3 points).  So what happens when we "cube" those three points, e.i., raise the points of that cube to the 3rd power?  We end up with a bigger cube in three dimensional space, just like we ended up with a bigger number in the one dimensional space example before.\n\nHowever, the answer to your question might possibly be 5 dimensions.  If you think of a number to the first power as being in the first dimension (on a line graph), and you cube it, it only moves up two dimensions, from the 1st to the 3rd (a cubic graph of space).  So taking a 3 dimensional number and cubing it would also move it up just two more dimensions from the 3rd to the 5th.  I can't think of any way to write that with a conceptually valid math expression, though.\n\nI'm not exactly sure how we'd get to 6 dimensions, unless you're thinking something like x^3*x^3 = x^(3+3) = x^6, but that doesn't seem reasonable to me, either.\n\nInstead, let's try this:\n3^1 = 3\n3^2 = 9, our previous number times our base number, 3.\n3^3 = 27, our previous number times our base number, 3.  So the cube of 3, our starting example, is 27.\n\nWhat's the cube of our cube (27)?  I.e., what's 27^3?\n27^1 = 27\n27^2 = 729 (27 times 27)\n27^3 = 19683 (729 times 27)\n\nSo for our final number, we want to find out how many "dimensions" it took to reach that, right?  So, what exponent of 3 does 19683 represent?  In other words, 3^x = 19683, what's x?  To find that out, we need to convert it to logarithmic form.  log_base3(19683) = x.  x = 9.  On a standard calculator, enter log(19683)/log(3) to find that out.  And you can check the answer by doing 3^9, which comes out to be 19683.  You can also follow that whole process using starting numbers other than 3.\n\nTo write it out more formally, (x^3)^3 = x^(3*3), by one of the properties of exponents, which = x^9.\n\nOr what if we did x^(3^3), which would be x^27.  27 "dimensions" seems far too excessive for the cube of a cube, to me.\n\nSo, my final answer is 9 "dimensions", though with the caveat that I think "dimensions" is misleading.
46479
Oh boy. I'm not sure that i can explain why the rules are what they are, but i know what the rules are, and i can tell you:\n\n1. 45-45-90\n -The two legs are equal\nLeg 1=a\nLeg 2=b\na=b\n -The hypotenuse is 2 times the square root of a (or b, since they are equal)\n\n2.30-60-90\n -The key to this is the small leg, "a". The hypotenuse is equal to 2a (2 times "a").\n -The large leg is equal to 3 times the square root of the small leg, "a".\n\nHope this helps!\n\n-incarnadyne49\n\nThe first guy is wrong, i'm sure!!! I would know, i am a calc student, and access to a geometry textbook!!
386357
Disagree.  You can have a square that has sides of 8 inches and another with sides of 3 inches.  They are similar because the angles are congruent, and the sides are in proportion, but they are NOT congruent figures.  (Same SIZE and shape).  If two polygons are congruent, then their sides are in a scale factor of 1:1 and their angles are congruent, thus they are similar also.
1072143
i just got done with geometry and i hated it i'm more of an algebra person.  but to survive geometry you have to take notes and study!  there are so many formulas its not even funny! ugh shapes we're not my best thing in the world. but ask questions when you don't understand because when you don't understand one thing you won't understand a lot of things that go with it.  i hate geometry proofs!!!!!!!!!   check ya later ♥
208438
There is a delightful article by the late Dr. John Kenny on that, back a few years ago, looking at the formulae for surface area and volumes of spheres; I'll see if I can find it for you--it's in my files SOMEWHERE--and give you the reference, and the formulae he generated. It turns out that they're not precisely the same as what you've got--but not terribly far from it.\n\nAs I said, I'll see if I can find it, and I'll get you the reference and the formulae.\n\nAbout 9pm CST...\n\nOK, here is the formulae, done the best I can on this thing...\n\n\nFor a sphere of dimension 2*k, radius R\n\nVolume: (pi^k)*(R^(2*k))/k!\n\nFor a sphere of dimension 2*k + 1 and radius R\n\nVolume:\n(2^(2*k+1))*k!*(pi^k)* (R^(2*k+1))/(2*k+1)!\n\n\nTo get the surface areas for each of these, all you do is differentiate with respect to R...\n\nAs I said, it was a little more complex than you thought--the surface areas come out a little bit differently for odd and even numbered dimensions.  I do not recall why, frankly.
374695
its so easy ,\n   \n    we know that Arithmetic mean of n no.s > geometric mean of n no.s\n  \n there for four no.s a b c d\n  its AM is (a+b+c+d)/4\n  and its GM is  (abcd)^1/4\n\n  there fore \n    AM > GM\n   ie\n   (a+b+c+d) >  4 . (abcd)^1/4\n  \n        HENCE PROVED ,\n   AND PLEASE DONT ASK SUCH STUPID QUESTIONS\n                !!!!!!  STUPID PEOPLE STUPID QUESTIONS !!!!!\n\nBECAUSE GENIUS LIKE EINSTEIN NAD NEWTO ARE STILL ALIVE IN THIS WORLD
664409
like .-1 said\n\nthat would make it, for example, the ratio of lengths of hypotenuse to either leg of an isoceles right triangle.\n\nor the ratio of the length of a diagonal of a square to the length of any of its sides.\n\nalso the factor to scale a plane figure in order to double its area\n\netc.
521925
You find it in the normal coordinates way; i.e.\n((x1+x2)/2, (y1+y2)/2)\n\nThe curvature of our Earth does not affect it as the coordinates we are using is 2-dimensional.
788027
The graph is a V.  From the 0,0 draw a line with slope = 1 to the right of the y axis and draw a line with slope = -1 to the right of the y axis.  For y<|x|.  All you have to do is shade in the area below the V.  If y>|X|, all you have to do is shade in the area above the V.  Less than, shade below; greater than, shade above.  That should answer all the questions you've submitted.\n\nAnd no, absolute values do not form parabolas.  Some do, not all.  Like y = sqrt[|x-1|].  This forms 2 graphs each going away from each other.\nNice going NerdyKid.  You're not a nerd, you're a dork.
164065
The trick with "proofs" like these are generally an obscured division by zero. One of the lines seems to work, though when you divide to cancel out some factors, the quantity you're dividing by could/does equal zero. It's a trick and doesn't say anything substantial about mathematics.
128613
Step 1) Make a list of all the distances you need to scale expressed in feet.\n\nStep 2) Examine the list and find the largest distance you need to scale. Let's call this distance L.\n\nStep 3) Divide 375 by L. This is the scaling factor. Let's call it S. It will be some quantity that is much less than 1. For example if the distance to Washington were the largest number on the list then the scaling factor would be 375/7336824=0.0000511120343080. To facilitate calculations, you should save this scaling factor into the memory of your calculator.\n\nStep 4) Multiply each distance on your list by S. This will produce a new list of numbers where the greatest distance has been scaled to 375 feet and all the other numbers are proportionately less than 375.\n\nHope this helps.
1064957
Do you have a small scale?  Put a magnet on the scale with the north pole up.  Mount another magnet above that magnet with the north pole down.  Move the top magnet to different heights and records the height and the "weight" on the scale.  Mount the second  magnet with the south pole down and repeat.  Draw a graph comparing height and weight with the north pole  up and the north pole down.  What shapes are the graphs?
244986
"Scale" refers to the ratio between the size of an object (or details of an object) and a representation of that object. So, when a map is "drawn to scale", it means the features of of the map are the same size relative to each other as the real objects they represent. Similarly, if you have a drawing of a person who is 6 feet tall, and has legs 3 feet in length, and the drawing is 6 inches tall, the legs in the drawing would be 3 inches in length if the drawing is to scale.
1062497
What everyone has said is correct, but look at it this way: Consider a square. We know from the theorem of pythagoras that the ratio of the hypotenuese to the length of one side = sqrt(2) or, if you like, the ratio of one side to the hypotenuese = 1/sqrt(2). These constant ratios are independent of the size of the square. We have the ratio of the perimeter of the square to the hypotenuese = 4/sqrt(2) = 2*sqrt(2), which is independent of the size of the square.  Similarly the ratio of the circumference of a given circle to the longest chord(the diameter) has an exact value, and similarly this value is independent of the size of the circle. As has been mentioned, for any regular n-gon(square, pentagon, hexagon etc) the ratio of the perimeter to the length of the longest line segment that can be drawn within the n-gon will have a specific value depending on n, and this value can be determined. As we let n approach infinity, the n-gon approaches a circle and the ratio of the perimeter of the n-gon to the longest "chord" approaches a specific limit and this limit is pi, the ratio of the circumference to the diameter.  And, as mentioned above, analytic techniques can (and have) determined the value of the constant pi to a trillion or so digits.\n\nFootnote: SK, determining the limit of a series or even that the limit exists does not necessarily require calculus.\nFor example, the sum of a geometric series such as Sum[1/(2^n)] = 1/2 + 1/4 + 1/8+....+1/2^n is proved without calculus to be 1- 1/2^n and the limit of it (without calculus) as n---> inf. = 1. Check out the below link regarding pi.\n\n http://mathworld.wolfram.com/ArchimedesAlgorithm.html
183218
a. = gradiant scale, because you can choose degrees (gradiants) between one pole and the other pole.\n\nb. = binary scale, because you only have two choices and can only select one of them\n\nc. = defined gradiant, because each of the degrees (gradiants) is pre-defined for you.\n\nI hope this is the best answer.
709666
Use a graph paper. Vector normally given in the form of (i,j). Right now, assume your graph x-axis is your value of 'i' and y-axis is your value of 'j' plot them in and you will get it.\n\nRemember to size your scale as you do it in a normal mathematical way so that your whole vector line will be on 1 piece of graph paper.