Sam and John have walked at right angles to each other, so the distance between them is the hypotenuse of a triangle.  The distance can be found using the Pythagorean Theorem.

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.

By drawing Angela’s route, we can connect her end point and her start point with a straight line and will then have a right triangle. The Pythagorean theorem can be used to solve for how far she is from the starting point: a²+b²=c², 30²+40²=c², c=50. It can also be noted that Angela’s route represents a multiple of the 3-4-5 Pythagorean triple.

Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: c² = 3²+ 4² = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.

This question requires the use of Pythagorean Theorem. We are given the length of two sides of a triangle and asked to find the third. We are told that the two sides we are given meet at a right angle, this means that the missing side is the hypotenuse. So we use a² + b² = c², plugging in the two known lengths for a and b. This yields an answer of 27.46 feet.

The Pythagorean theorem is a² + b² = c², where a and b are legs of the right triangle, and c is the hypotenuse.

(15)² + (20)² = c² so c² = 625.  Take the square root to get c = 25m

By using the Pythagorean Theorem, we can solve for the distance “as the crow flies” from Paul to his home:

3² + 4² = x²

9 + 16 = x²

25 = x²

5 = x