You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of , what will the length of the hypotenuse of the triangles be?

This problem is trying to distract you by thinking of triangles. What we are really asked to find here is the length of the diagonal of a square with sides of 15 inches.

Splitting a square along its diagonal yields two 45/45/90 triangles. If you know the ratios for 45/45/90 triangles, you can find the answer very quickly.

Think:

Meaning that if the two short sides are x units long, the hypotenuse will be x times the square root of two units long.

In our current case, our short sides are 15 inches long, so our hypotenuse will be 

You could also solve this with Pythagorean Theorem.

a and b are both 15 in, so we can solve.

So,our answer is 

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?

To find the diagonal of a square, we can recognize one of two things.

1) The diagonal of a square creates a right triangle, and we can use Pythagorean theorem to find our diagonal.

2) The diagonal of a square creates two 45/45/90 triangles, with side length ratios of 

Using 2), we can find that the diagonal of the square must be 

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?

So, we need to find the diagonal of a square. First, we need to find the side length.

Let's begin with our formula for the area of a square:

where s is our side length and A is our area.

With this formula, we can solve for our side length by plugging in our area and square rooting both sides.

Now, to find the diagonal, we can think of an isosceles right triangle, where the two equal sides are 15 ft. This is also a 45/45/90 triangle, which means the side lengths follow the ratio of  . 

This means our answer is    .

We could also find this by using Pythagorean Theorem.

Area of a quadrilateral is found by length times width. In a square, these are the same, so the length of side   is a number that, when multiplied by itself is equal to 169.

In other words, take the square root of 169 to find the length of .

To find the perimeter of a square, we will use the following formula:

where a is the length of any side of the square.  Because a square has 4 equal sides, we can use any side in the formula.  To find the length of any side, we will solve for a.  

Now, we know the perimeter of the square is 36in.  So, we will substitute. 

Therefore, the length of one side of the square is 9in.

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?

Let's begin with our formula for the area of a square:

where s is our side length and A is our area.

With this formula, we can solve for our side length by plugging in our area and square rooting both sides.

A square has 4 equal sides.  The formula to find perimeter of a square is:

where a is the length of any side.  Now, to find the length of one side, we will solve for a.

We know the perimeter of the square is 60cm.  So, we can substitute and solve for a.  We get

Therefore, the length of one side of the square is 15cm.

Like any polygon, the perimeter of a trapezoid is the total distance around the outside, i.e.

,

where are the lengths of each side.

Let small base length and big base length. 

The perimeter of a trapezoid is equal to the sum of all of the sides, i.e.

,

where are the lengths of each side. 

Substitute  in the formula for the area of a trapezoid: