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The Pythagorean Theorem on the GMAT

There’s a reason this is the most famous theorem in mathematics!  This remarkable theorem is one of the most versatile and highly adaptable formulas in existence.  Of course, I’m sure you remember that it says: For any right triangle,

 
 
 
 
 
 
 
 
 
 
 

Of course, if any question gives you two sides of a right triangle and asks you to find the third, you will use this formula. Here are a couple problems to show its other guises.

 

 

Practice Questions: Using the Pythagorean Theorem

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

1) In the coordinate plane, point A has coordinates (-2, -1) and point B has coordinates (4, 7). What is the length of segment AB?

    (A) 10
    (B) 10.5
    (C) 12
    (D) 13.675
    (E) 14

2) Paul drove 50 miles north, then changed direction and drove 120 miles east. At the end of this trip, how far was he from his starting point?

    (A) 70 miles
    (B) 110 miles
    (C) 130 miles
    (D) 150 miles
    (E) 170 miles

 

Answers and Explanations

1) The easiest distances to find in the coordinate plane are the horizontal and vertical lines, so we begin by drawing those.

For the lengths of AC and BC, we can simply count boxes. AC = 6 and BC = 8. These are the two “legs” of a right triangle, and AB is the hypotenuse, so

(AC)^2+(BC)^2=(AB)^2

(6)^2+(8)^2=(AB)2

36+64=100=(AB)^2

AB=10

Answer: A.

 

2) Essentially, Paul drove along the legs of a big right triangle, and the hypotenuse is how far he is from his starting point.

Therefore, (distance from start)^2=50^2+120^2=2500+14400=16900 right

distance from start =sqrt{16900}=130

Answer: C.

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5 Responses to The Pythagorean Theorem on the GMAT

  1. Sirisha December 2, 2014 at 8:27 am #

    Hi, In the explanation to problem 1 above, you mentioned to simply count the number of boxes to get AC and BC. In the real GMAT, is the co-ordinate plane given in box style? Thank you.

    • Mike MᶜGarry
      Mike December 2, 2014 at 10:13 am #

      Dear Sirisha,
      I’m happy to respond. 🙂 Sometimes the GMAT gives the gridlines and sometimes it doesn’t, but even if you are given no diagram at all, it should not be hard to figure out, say, that from the x-coordinate of A, -2, to the x-coordinate of B, 4, there are 6 units. You shouldn’t need the picture at all to be able to count the number of units along a horizontal or vertical line in the coordinate plane.
      Does this make sense?
      Mike 🙂

      • John Green March 21, 2017 at 7:53 pm #

        If it only gives coordinates of ‘A’ and ‘B’… what’s the best way for me to know that it is indeed a right triangle? I’m visually oriented so sometimes seeing text is hard for me to decipher into a picture into my mind.

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert March 21, 2017 at 11:42 pm #

          In many questions, including the second question in this post, you’ll be explicitly told that the triangle is a right triangle. In that case, just look at the angle that clearly looks “right” (visually like the corner of a square), and you’ll have the visual cue that helps you really store that image in your mind.

          In the case of the first question, every square in a coordinate plane grid is a set of right angles. (Because, of course, squares have four right angles.) So simply envision the grid boundaries as part of the right triangle, and you’ll be able to visualize your right triangle.

          Still in other cases, the problem will tell you that you have a right triangle without using text. Instead, the “right angle” symbol will appear within the triangle figure. This symbol is a little square inside the right angle. To see what this looks like, check out the main image (top, right hand corner) for the Wikipedia article on “right triangle.

    • SSY February 6, 2015 at 6:32 pm #

      yes, sometimes it does and sometimes it doesnt.


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