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mathmath333
 one year ago
Prove
mathmath333
 one year ago
Prove

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} &\text{Prove that area of the figure}\hspace{.33em}\\~\\ &x+y=k ,\ k>0,\ k\in \mathbb{R}\hspace{.33em}\\~\\ &\text{is same as area of figure }\hspace{.33em}\\~\\ &x+a+y+b=k,\ \{a,b\}\in \mathbb{R}\hspace{.33em}\\~\\ \end{align}}\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Almost the same as the previous one  try to figure out the four lines.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} x+y=k\hspace{1.5em}\\~\\ xy=k\hspace{1.5em}\\~\\ x+y=k\hspace{1.5em}\\~\\ xy=k\hspace{1.5em}\\~\\ \end{align}}\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Yeah. Now graph them and see what you get.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1square with area \(k^2\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4Same process for the second case.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1what about 'a' and 'b'

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4don't worry about them. figure out the equations first.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4looks like the area is not \(k^2\) though.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} x+ayb=k\hspace{1.5em}\\~\\ x+a+y+b=k\hspace{1.5em}\\~\\ xa+y+b=k\hspace{1.5em}\\~\\ xayb=k\hspace{1.5em}\\~\\ \end{align}}\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4the diagonal of the square in the first case is \(2k\), so the side length is \(k\sqrt 2\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4yeah, can you manage the rest now?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1area of the first is \(2k^2\) ?

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1and how to find 2nd one

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4just find out the points where the lines meet and those would be the vertices of the square.

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3hint: if we make this traslation: \[\large \left\{ \begin{gathered} x + a = X \hfill \\ y + b = Y \hfill \\ \end{gathered} \right.\] where X,Y are the new coordinates, then the second equation can be rewritten as follows: \[\large \left X \right + \left Y \right = k\] and such equation has the same shape of the first one

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1would this be enough for proof

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3Yes I think so, since we have applied a traslation, and not a dilation, namely when we traslate a geometrical shape its area will be unchanged

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.4That is actually a nice way to see it, but it seems that he actually has to *prove* that translation has no effect on the shape, and in turn, has no effect on area.

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.3I think that a proof of my statement above, can be this: when we traslate a rigid body into the euclidean space, its volume is unchanged, now we can imagine to build the geometrical shape involved into our exercise using rigid rods, so we can conclude that the area enclosed by our geometrical shape will be unchanged after a traslation
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