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GMAT Math: Special Properties of the Line y = x

The 45º angle

Fact: All lines with slopes of 1 make 45º angles with both the x- and y-axes.

Conversely, if a line makes a 45º angles with either the x- of y-axes, you know immediately its slope must be pm 1. This first fact is true, not only for y = x and y = –x, for all lines of the form y = mx + b in which m equals either 1 or –1.  If the slope is anything other than pm 1, you would need trigonometry to figure out the angles, and that’s beyond the scope of GMAT math.  The GMAT could expect you to know this one fact about these special lines, especially on Data Sufficiency.

 

As a Mirror

Fact: Suppose we treat the line y = x as a mirror line.   If you take any point (a, b) in the coordinate plane, and reflect it over the line y = x, the result is (b, a).  It reverses the x- and y-coordinates!

The corollary of this is that if we compare any two points with reversed coordinates, say (2, 7) and (7, 2), we automatically know that each is the image of the other by reflection over the line y = x.  Add now the geometry fact that a mirror line is the set of all points equidistant from the original point and its image.  This means that the midpoint of the segment connect (2, 7) and (7, 2) must lie on the line y = x.  In fact, any point on the line y = x will be equidistant from both (2, 7) and (7, 2).  Without doing a single calculation, we know, for example, that the triangle formed by, say, (2, 7) and (7, 2) and (8, 8 ) must be an isosceles triangle. (See the diagram below.)

When we reflex over the line y = –x, the coordinate are reversed and made their opposite sign: e.g. (2, 7) reflect to (–7, –2), and (–5, 3) reflects to (–3, 5).  The other conclusions, about equidistance, remain the same.

 

As a Boundary

Fact: Any point (x, y) in the coordinate plane that is above the line y = x has the property that y > x.  Any point (x, y) in the coordinate plane that is below the line y = x has the property that y < x.

Can you sense the veritable cornucopia of Data Sufficiency questions that could arise from this fact?  If you every see a question about the coordinate plane asking whether y > x or y < x, chances are very good that the line y = x is hidden somewhere in the question.

 

Practice Questions

1) Is the slope of Line 1 positive?

Statement #1: The angle between Line 1 and Line 2 is 40º.
Statement #2: Line 2 has a slope of 1.

(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D) Each statement alone is sufficient to answer the question

(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

 

2) Point (P, Q) is in the coordinate plane.  Is P > Q?

Statement #1: P is positive.
Statement #2: Point (P, Q) above on the line y = x + 1

(A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

(B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

(C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

(D) Each statement alone is sufficient to answer the question.

(E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

 

3) A circle has a center at P = (–4, 4) and passes through the point (2, 3).  Through which of the following must the circle also pass?

(A) (1, 1)

(B) (1, 7)

(C) (–1, 9)

(D) (–3, –2)

(E) (–9, 1)

 isosceles triangle xy plane

Practice Questions Explanations

1) A straightforward prompt.

Statement #1 is intriguing: it gives us a specific angle measure.  This is tantalizing, but unfortunately, it is only the angle between Line 1 and Line 2, and that angle could be oriented in any direction.  Therefore, we can draw no conclusion about the prompt from this statement alone.  Statement #1, by itself, is insufficient.

Statement #2 is also tantalizing, because it’s numerically specific.  But, unfortunately, this tells us a lot about Line 2 and zilch about Line one, so this statement is, by itself, is also insufficient.

Now, combine the statements.  From statement #2, we know Line 2 has a slope of 1, which means the angle between Line 2 and the positive x-axis is 45º.   We know, from statement #1, that Line #1 is 40º away from Line 2.  We don’t know which way, above or below Line 2.  If Line 1 is steeper than Line 2, it makes an angle of 45º + 40º = 85º with the positive x-axis.  If Line 1 is less steep than Line 2, it makes an angle of 45º – 40º = 5º with the positive x-axis.  Either way, its angle above the positive x-axis is between 0º and 90º, which means it has a positive slope.  The combined statements allow us to give a definitive answer to the prompt question.  Answer = C.

 

2) We see the x > y type question in the prompt, which makes us suspect that the line y = x will play an important part at some point.

Statement #1 just tells us P is positive, nothing else.  The point (P, Q) = (4, 2) has the property that P > Q, but the point (P, Q) = (4, 5) has the property that P < Q.  Clearly, just knowing P is positive does nothing to help us figure out whether P > Q.  Statement #1, by itself, is wildly insufficient.

Statement #2 is intriguing.  It discusses not the line y = x but the line y = x + 1.  What is the relationship of those two lines?  First of all, they are parallel: they have the same slope.  The line y = x has a y-intercept of zero (it goes through the origin), while the line y = x + 1 has a y-intercept of 1.  This means: any point on the line y = x + 1 must be above the line y = x.  If (P, Q) is on y = x + 1, then it is above y = x, which automatically means Q > P.  We can give a definite “no” answer to the question.  By itself, Statement #2 is sufficient.  Answer = B.

 

3) For this problem, there’s a long tedious way to slog through the problem, and there’s a slick elegant method that gets to the answer in a lightning fast manner.

The long slogging approach — first, calculate the distance from (–4, 4) to (2, 3).  As it happens, that distance, the radius, equals sqrt{37} .  Then, we have to calculate the distance from  (–4, 4) to each of the five answer choices, and find which one has also has a distance of   sqrt{37} —- all without a calculator. 🙁

The slick elegant approach is as follows.  The point (–4, 4) is on the line y = –x, so it is equidistant from any point and that point’s reflection over the line y = –x.  The reflection of (2, 3) over the line y = –x is (–3, –2).  Since (–3, –2) is the same distance from (–4, 4) as is (2, 3), it must also be on the circle.  Answer = D.

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27 Responses to GMAT Math: Special Properties of the Line y = x

  1. Mukhtar November 12, 2016 at 10:45 am #

    hello guys thanks for your great job
    I have a question about practice number 3….Do I have to do all of this calculations because when I saw the point (4.-4) I knew immediately that it’s a y=-x line so what I did is just reversing the points and the signs of the point (2,3).

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert November 16, 2016 at 8:17 am #

      That is the quick “number sense” way to solve the problem. Good job recognizing the y = x aspect of this problem and finding the “reflection” of your given (2,3) coordinate.

  2. Gaurav Sharma September 23, 2016 at 11:26 pm #

    Hi Mike,

    Thanks for the great post.

    I’ve one doubt, in statement above w.r.t y = -x

    When we reflex over the line y = –x, the coordinate are reversed and made their opposite sign: e.g. (2, 7) reflect to (–7, –2), and (–5, 3) reflects to (–3, 5). The other conclusions, about equidistance, remain the same.

    We are saying that the co-ordinates are reversed and the made their opposite signs:

    1. (2, 7) mirror image will have following co-ordinates (-7, -2)

    2. Whereas (-5, 3) mirror image will have following co-ordinates (-3, 5)

    In second case we are reversing the co-ordinates but not the signs? Which is contradicting the above statement.

    Pls help.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert October 1, 2016 at 8:38 pm #

      Hi Gaurav,

      So, remember that, “when we reflex over the line y = –x, (1) the coordinate are reversed and (2) made their opposite sign.” Now, let’s take a closer look at the second case: (-5, 3).

      Step 1: Reverse the coordinates
      (-5, 3) >>>> (3, -5)

      Step 2: Coordinates have opposite signs
      (3, -5) >>>> (-3, 5)

      I hope this helps!

  3. abdul June 18, 2016 at 2:37 am #

    Mike,
    I have learned more from you than what I have learned from my teachers in high school, bachelors,masters, and PhD. Am an aspiring mathematician,and these little concepts in your blog posts give my brain a good workout before i work on denser theories.also, There was this Amelia token problem that really tricked me, and I had a good laugh;You can’t let your attention slip -no matter how beautiful a woman passes by you- especially when solving data sufficiency. On another post ,you mentioned what the audience thinks about number- picking vs algebra. I personally feel its intuitive. After doing ten thousand problems, one often develops an intuition that tells him/her whether to pick numbers or use algebra. To be honest, one can just pick numbers and still do very well on an exam like GMAT. I always look forward to yours and karismas’ blogs and been doing it for many years. I think its high time that magoosh should open a University that rivals the universities that are in existence only to serve the elites. If magoosh plans to do so, don’t forget me a job offer:)Moreover, I love it when you discuss roger Perry’s, left-brain vs right – brain theory. Yes, we can definitely sharpen our analytical and creative minds. There are fast tracks to do them but I don’t want to discuss on this public blog and corrupt our youths innocent minds. Proper sleep,hydration,nutrition,meditation/mindfulness is enough for learning anything. Nevertheless, at the end of the day, GMAT is not a reasoning, iq or logic test. It’s just a test of working memory. magoosh problem sets do juice up the neurons and synapses in the brain, thereby increasing the ram.

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 18, 2016 at 3:29 am #

      Hi Abdul,

      Thanks for the kind words! I think we are a long way from becoming Magoosh University, but it is very gratifying to know that you value our approach and style that way! 🙂

      You have a lot of great points here, and it is fantastic to know that our blog posts continue to stimulate your (very academically qualified!) brain. May your neurons and synapses be ever active! 😀

    • Mayank June 22, 2016 at 6:59 am #

      From the nuances of the English Grammer to the Nerve -wracking Quant !!
      Mike – You are absolutely Amazing.
      By the way I follow your blogs quite regularly, and they are truly insightful !

      • Magoosh Test Prep Expert
        Magoosh Test Prep Expert June 22, 2016 at 1:24 pm #

        Thanks for the kind words, Mayank! 🙂

  4. amit November 29, 2015 at 5:45 pm #

    mirror image of point (5,10) about a line makes an angle of 135 with positive x axis is this follow y=-x

  5. Ananth October 9, 2014 at 11:13 am #

    Dear Mike;

    I didn’t understand the answer to Question #2

    The explanation says B. but what if we consider Quadrant 3 in which the line Y=X+1
    extends. then Q being above P would always be < P since both Q & P would be negative values.

    In that case the Statement 1 fact 'P being positive' becomes mandatory right ?

    So, shouldn't the answer be C ?

    Please lemme know quick ! GMAT in 2 days !!!

    • Mike MᶜGarry
      Mike October 9, 2014 at 11:53 am #

      Dear Ananth,
      I’m happy to respond. 🙂 My friend, I think you are getting confused on how “greater than” and “less than” work with negative numbers. A negative number with a smaller absolute value is actually greater than a negative number with a larger absolute value. For example, (-10) > (-100). Think about it: who is in better financial shape, the person with $10 of debt, or the person with $100 of debt? Less debt is better, and in much the same way, (-10) is greater than (-100) — (-10) is to the right of (-100) on the number line, in precisely the same way that (+100) is to the right of (+10) on the number line.
      Suppose we are in QIII. Suppose P = -3. Then, the point on the line y = x + 1 would be (-3, -2), and (P, Q) is above this, so Q could be -1, 0, or some positive number. All of those are greater than -3. Zero is greater than any negative number. Again, one is better financial shape with zero debt than with any non-zero amount of debt.
      Does all this make sense?
      Mike 🙂

  6. Tushar March 31, 2014 at 3:19 pm #

    cool post, mike.

    • Mike MᶜGarry
      Mike March 31, 2014 at 3:51 pm #

      Dear Tushar,
      I’m glad you like it. Best of luck to you!
      Mike 🙂

  7. Sandy Nguyen November 28, 2013 at 9:47 pm #

    A really well-written post about an interesting subject. Thank you so much!

    • Mike MᶜGarry
      Mike November 29, 2013 at 10:45 am #

      Sandy,
      You are quite welcome. Best of luck to you.
      Mike 🙂

  8. Piyush Jain March 18, 2013 at 11:43 pm #

    Dear Mike

    With reference to the Mirror system mentioned above, can we comment on the distance between suppose (2,3) & (3,2) with respect to line y = -x ?

    • Mike MᶜGarry
      Mike March 19, 2013 at 10:26 am #

      PIyish
      Any point on the line y = x is equidistant to (2, 3) & (3, 2), but there is no particular symmetry statement possible for a general point on y = -x and these two points. I would strongly suggest drawing an accurate graph and investigating this visually for yourself.
      Mike 🙂

      • Piyush Jain March 19, 2013 at 10:54 am #

        Thank you

        • Mike MᶜGarry
          Mike March 19, 2013 at 12:03 pm #

          You’re welcome.
          Mike 🙂

      • Arun Panda December 11, 2014 at 11:04 pm #

        “With reference to the Mirror system mentioned above, can we comment on the distance between suppose (2,3) & (3,2) with respect to line y = -x ? ”

        The line joining points (2,3) and (3,2) will be parallel to line y = -x.

        Kindly comment on my interpretation.

        • Mike MᶜGarry
          Mike December 12, 2014 at 10:36 am #

          Dear Arun,
          Yes. The line between a point and its reflection is always perpendicular to the mirror line. If we reflect over y = x, the line between the two points has to be perpendicular to y = x. The line y = -x is perpendicular to the line y = x, and two lines perpendicular to the same thing must be parallel to each other.
          Does all this make sense?
          Mike 🙂

  9. Piyush Jain March 18, 2013 at 9:00 am #

    The mirror trick really saves a lot of time, specially for questions like the last one. Thank you.

    Also, there are limited posts on COORDINATE GEOMETRY and GRAPHS AND CHARTS and will really appreciate if you could throw in some more posts on these topics.

    Thank you

  10. Andy October 28, 2012 at 7:35 pm #

    Awesome post on the y=x concept !

    • Mike MᶜGarry
      Mike October 29, 2012 at 8:39 pm #

      Andy
      I’m glad you found it helpful. Best of luck to you!
      Mike 🙂

  11. KM July 5, 2012 at 8:12 am #

    Informative article, Mike !
    Thanks!

    • Mike MᶜGarry
      Mike July 5, 2012 at 9:17 am #

      Thank you for your kind words. I’m glad you found it helpful.
      Mike 🙂


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