Question

If N is an integer, wich of the following must be a rational number?
1. The ratio of the length of the diagonal of a square to the length of its side if the side is N centimeters
2. The ratio of the circumference of a circle to its diameter if its diameter is N centimeters
3. The ratio of the area of an equilateral triangle to its perimeter if its side is N centimeters
4. The ratio of the area of a circle to its circumference if its radius is N centimeters. Explain

Answer 1

You can start by thinking about what a rational number is.

One way of showing that somethiing is a rational number is by showing you can write it as a/b, where a and b are both integers.

Another way of showing that something is a rational number is seeing if the decimal expansion of the number ever stops or repeats. For example, 4.25 is rational because the digits stop. Or, if you have 3.6666.... with 6's forever afterward, that's a rational number because 6 repeats over and over again. Or 4.19191919... with "19" repeating forever, would also be rational.

So, you can think about whether pi and different square roots are rational numbers. (Some square roots are, some aren't.)

Answer 2

And th answer is?

Answer 3

Well, you first have to figure out what the ratio is in each of the 4 parts. So for part 1, you can find:

ratio = (square diagonal) / (square side) = (square diagonal) / N

you could use 45-45-90 triangles to find that.

Answer 4

Thanx foldnggirl but I need more explications to find the answer

Answer 5

Ok, do you see how the diagonal of a square cuts the square into two 45-45-90 triangles?

That means, if the side of the square is N, then the diagonal has to be \[Nsqrt{2}\]

Answer 6

i mean:

\[N \sqrt{2}\]

Answer 7

then the ratio in part (1) is:

\[N \sqrt{2} / N\] =
\[\sqrt{2}\]

Answer 8

so then the question really is:

is the square root of 2, a rational number?

Answer 9

Gotta go get something to eat, be back later...

Answer 10

No is not