area of square problem

Question

On square dot paper, draw a square with an area of 2 units^2. Write to explain how you know the square does have this area.

 

in this question again i couldnt use pythagoras theorem , then with out using pythagoras theorem how can i tell that the area of enclosing square will be 4 unit^2 and area of 1 triangle will be 1/2

any other method rather than pythagoras theorem can i use

please help me

Answer 1

You are using squared graph paper, right? When you have drawn a 2x2 square you just count how many squares of graph paper are enclosed. If the side of the square is 2 then the area enclosed is 4 squares.

But how do you draw a square with area 2? What you do is draw the diagonals of each of the 4 enclosed squares. For the top left and bottom right squares you draw the diagonal from the bottom left corner to the top right corner; and for the other two squares you draw the other diagonal. This gives you a square inside the bigger square tilted by 45 degrees.

You have also divided the area of 4 squares into 8 triangles. So 8 triangles have a total area of 4 little squares. Each triangle has an area of ½. The tilted square contains 4 of these triangles. So if 8 triangles is equivalent to 4, then 4 triangles is equivalent to 2. That's how you know the area of your tilted square is 2. So its sides have length √2, the length of each diagonal. No Pythagoras!

Looking at squares geometrically probably also helps you to answer your other question, when you start with the tilted square. Because the area of the tilted square is 2, its side length is √2. But the side is the diagonal of a unit square. This means to get the length of the side of the square knowing the length of its diagonal, you just divide the length of the diagonal by √2.

Comments

like that ? 

got it thank u so much sir , but couldnt understand 2nd question . here diagonal is the side of a unit square that make me confuse :(

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can i use 4 X 4 square ? or should i use only 2 X 2 ??

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A 2x2 square is sufficient, but the main thing is that you have shown the tilted square correctly made up of triangles that each have an area of ½. The tilted square shows the 4 triangles so you have shown that the tilted square has an area of 4*½=2.

Reverse the argument so that you're starting with the diagonals, which must each have a length of √2, because by definition, if area=(side)^2=2, then side=√2. You already know that this is the diagonal of a unit square, side=1, so the ratio of side:diagonal is 1:√2, or 1/√2:1, just by dividing the ratio by √2. You can also write 1/√2 as √2/2. The side is roughly 71% the length of the diagonal.