Math extra credit?

Question

If the vertcies of square "A" are the midpoints of the edges of square"B", then what is the ratio of the area of square "A" to square "B"?

Answer 1

Draw it out.
The smaller square should be rotated 45 degrees to the larger square.
Now draw diagonals between the corners of the smaller square.

You should have a square with 8 equal triangles. As you can see, 4 are within the smaller square. Therefore the area of the smaller square is 1/2 the larger.

Ratio: 4:8 or 1:2...

If you need a mathematical solution split the smaller square into 2 triangles, and solve for area based on the lengths of the sides of the larger square:
i.e.
Side of B = x
Area of B = x^2
Area of triangle = 1/2 b * h; b = x; h = 1/2 x
Area of A's 2 triangles = 2 * 1/2 * x * 1/2 x
Area of A = 1/2 x^2

Ratio is then 1:2, clearly not 1:8...

Answer 2

First, draw one square and mard the midpoints of the edges. Inscribe a second square using the midpoints as the vertices, then you have a visual representation of the problem.

Assume, for instance that the outer square had a length of 2. A triangle formed by the length of a side of the inner square will have two sides equal to 1. Using the Pythagorean theorem, you find that the side of the inner square is of the form 1^2 + 1^2 = 2^2, or 1.414. You ought to be able to figure out the ratio from there.

--Dee

Answer 3

1:2

Answer 4

Draw a picture. Then draw the diagonals of the small square. Now count triangles to get the ratio of areas.

Alternatively, read a dialogue of Plato's. IIRC, the Meno.

Answer 5

The same as the ratio of your the space your brain takes up to the space inside your skull: 1:2

Answer 6

1:8
by drawing a figure we get the ans
if the side of square A is a
then side of square B is a root2

the area of first square is a^2
area of second one is 8a^2

Answer 7

ratio of sides=2
ratio of areas=4
ratio of volumes=8