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 .

 cannot be the side lengths of a right triangle.  does not equal . Also, special right triangle  and  rules can eliminate all the other choices.

It is easiest to solve this using special triangle formulas. From the description, we know this is a right triangle with legs of 10 and 24. 10 and 24 are multiples of the special triangle with side ratio 5 :12 :13. 

Since we multiplied 5 and 12 by 2 to get 10 and 24, respectively, we compute the diagonal to be 2 * 13 = 26.

Using the Pythagorean Theorem, a² + b² = c², is also an acceptable approach to solving this problem.

These two triangles are similar triangles.  The triangle on the right is one-third the size of the triangle on the left.  We know this by comparing the bottom legs of each triangle.

3/9 = 1/3 

Given this, all the other legs of the triangle on the right will be 1/3 the size of their corresponding legs on the left.

12 * 1/3 = 4
18 * 1/3 = 6

Now we know the lengths of the three sides, we simply add them together to receive our answer.

4 + 6 + 3 = 13

An alternate way to do this problem would be to take 1/3 the perimeter of the triangle on the left.

12 + 18 + 9 = 39

1/3 * (39) = 13

Either method will enable you to arrive at the same solution.

These three points form a right triangle on the Cartesian coordinate system. The perimeter is comprised of the length from ; the length from ; and the length of the hypotenuse between .

The perimeter is .