GRE Math : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #151 : Plane Geometry

Which of these triangles cannot have a right angle?

Possible Answers:

5, 12, 13

2, 2, 2√2

9, 12, 15

6,7,12

√2, √2, 2

Correct answer:

6,7,12

Explanation:

6, 7, 12 cannot be the side lengths of a right triangle. 62+ 7does not equal 122. Also, special right triangle 3-4-5, 5-12-13, and 45-45-90 rules can eliminate all the other choices.

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Daria and Ashley start at the same spot and walk their two dogs to the park, taking different routes. Daria walks 1 mile north and then 1 mile east. Ashley walks her dog on a path going northeast that leads directly to the park. How much further does Daria walk than Ashley?

Possible Answers:

2 – √2 miles

2 + √2 miles

√2 miles

1 mile

Cannot be determined

Correct answer:

2 – √2 miles

Explanation:

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.

Example Question #2 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Which of the following sets of sides cannnot belong to a right triangle?

Possible Answers:

3, 4 ,5

6, 7, 8

2, 2√3, 4

5, 12, 13

2, 2, 2√2

Correct answer:

6, 7, 8

Explanation:

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.

Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Max starts at Point A and travels 6 miles north to Point B and then 4 miles east to Point C. What is the shortest distance from Point A to Point C?

Possible Answers:

5 miles

4√2 miles

7 miles

2√13 miles

10 miles

Correct answer:

2√13 miles

Explanation:

This can be solved with the Pythagorean Theorem.  

62 + 42 = c2

52 = c2

c = √52 = 2√13

Example Question #161 : Plane Geometry

Which set of side lengths CANNOT correspond to a right triangle?

Possible Answers:

7, 24, 25

5, 12, 13

6, 8, 11

3, 4, 5

8, 15, 17

Correct answer:

6, 8, 11

Explanation:

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 = 102.

Example Question #31 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Kathy and Jill are travelling from their home to the same destination. Kathy travels due east and then after travelling 6 miles turns and travels 8 miles due north. Jill travels directly from her home to the destination. How miles does Jill travel? 

Possible Answers:

\dpi{100} \small 16\ miles

\dpi{100} \small 10\ miles

\dpi{100} \small 14\ miles

\dpi{100} \small 12\ miles

\dpi{100} \small 8\ miles

Correct answer:

\dpi{100} \small 10\ miles

Explanation:

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

  \dpi{100} \small 6^{2}+8^{2}=x^{2}

\dpi{100} \small 36+64=x^{2} 

\dpi{100} \small x=10 miles

Example Question #131 : Geometry

Square  is on the coordinate plane, and each side of the square is parallel to either the -axis or -axis. Point  has coordinates and point  has the coordinates .

Quantity A:  5\sqrt{2}

Quantity B: The distance between points  and

Possible Answers:

Quantity B is greater.

 

The two quantities are equal.

 

Quantity A is greater.

 

The relationship cannot be determined from the information provided.

 

Correct answer:

The two quantities are equal.

 

Explanation:

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 5\sqrt{2}.

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