The ratio of the sides of a 30-60-90 triangle is , with the hypotenuse being . Thus, the perimeter of this triangle would be . Since the triangle depicted in this problem has a perimeter of ,  must equal 1, which would make the hypotenuse equal to 2.

Use the Pythagorean theorem,   , with   and  , and solve for .

Rearrange to isolate :

Use FOIL to multiply out:

Distribute the minus sign to rewrite without parentheses:

Combine like terms:

Take the square root of both sides:

At first sight, it's tempting to assume this is a right triangle and to thus use the Pythagorean Theorem to find a length of 5 for the missing side.

However, the triangle was not stated to be a right triangle in the problem statement, and no indication was given in the drawing to indicate that it was a right triangle either, such as a square demarcation in the vertex opposite the side measuring 13.

Thus there is not enough information to give the length of the missing side. When taking standardized math tests, be careful making assumptions about information that is not given.

If an equilateral triangle is divided in 2, it forms two 30-60-90 triangles. Therefore, the side of the equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle. The side lengths of a 30-60-90 triangle adhere to the ratio x: x√3 :2x. since we know the hypothesis is 4, we also know that the base is 2 and the height is  2√3. 

First let's calculate how far Daria walks. This is simply 1 mile north + 1 mile east = 2 miles. Now let's calculate how far Ashley walks. We can think of this problem using a right triangle. The two legs of the triangle are the 1 mile north and 1 mile east, and Ashley's distance is the diagonal. Using the Pythagorean Theorem we calculate the diagonal as √(1² + 1²) = √2. So Daria walked 2 miles, and Ashley walked √2 miles. Therefore the difference is simply 2 – √2 miles.

To answer this question without plugging all five answer choices in to the Pythagorean Theorem (which takes too long on the GRE), we can use special triangle formulas. Remember that 45-45-90 triangles have lengths of x, x, x√2.  Similarly, 30-60-90 triangles have lengths x, x√3, 2x.  We should also recall that 3,4,5 and 5,12,13 are special right triangles.  Therefore the set of sides that doesn't fit any of these rules is 6, 7, 8.

This can be solved with the Pythagorean Theorem.  

6² + 4² = c²

52 = c²

c = √52 = 2√13

Because we are told this is a right triangle, we can use the Pythagorean Theorem, a² + b² = c². You may also remember some of these as special right triangles that are good to memorize, such as 3, 4, 5.  

Here, 6, 8, 11 will not be the sides to a right triangle because 6² + 8² = 10^2.

Kathy's path traces the outline of a right triangle with legs of 6 and 8. By using the Pythagorean Theorem

miles

To find the distance between points  and , split the square into two 45-45-90 triangles and find the hypotenuse. The side ratio of the 45-45-90 triangle is , so if the sides have a length of 5, the hypotenuse must be .