Isometries and Coordinate Geometry

Michael Pershan’s post on teaching congruence reminded me that way back in the beginning of summer, I’d been meaning to write up some of my geometry work, which I think is pretty unusual. Still on the list is the lesson sequencing, but here is some thoughts and sample problems on integrating Isometries and coordinate geometry.

To summarize my earlier work, explicated in Teaching Congruence, or Are You Happy, Professor Wu?, I was unhappy with the circular reasoning that geometry books present in congruence sections. Triangle ABC is congruent with Triangle DEF because all their sides and angles are congruent, and congruence is when the shapes have congruent sides and angles. Professor Wu’s writing taught me the link between congruence, similarities, and isometries (aka, transformations, or translations, rotations, and reflections). I’d previously skipped isometries, since the kids don’t need them much and they’re easy to figure out, but this discovery led me to use isometries as an introduction to congruence and similarity.

But all book chapters on isometries are very thin, or they rely on non-coordinate shapes, which is largely a waste of time. Was there any way I could bring back some other concept while working with isometries, particularly with my top students?

Which leads me straight to coordinate geometry. The most immediate tie-in is helping students figure out rotation, the most difficult of the transformations. A 90 degree rotation around a point involves perpendicular lines (“..and class, what is the relationship of perpendicular slopes, again? Class? Waiting!”). Moreover, the kids learn that the slope of the line connecting a point and its reflection must be perpendicular to the line of reflection. Finally, dilations involve all sorts of work with parallel lines. All of these reinforcements are excellent for weaker students, and are yet another reason to introduce transformations, even if only as a prelude to congruence.

But I wanted a meaningful connection for my top students, who usually grasp the basics quickly. What could I give them that would integrate algebra, coordinate geometry, and a better understanding of transformations?

Over the summer, I taught an enrichment geometry class to seventh graders whose parents got mad because I wasn’t assigning enough homework. My boss backed me–thanks, boss!—and the kids did, too—thanks, kids!—and not for the usual reasons (these are not kids who celebrate a lack of homework). The kids all told the boss that they were surprised that they weren’t able to just follow the pattern and churn out 50 problems of increasing difficulty in the same vein. “I have to really think about the problem,” said more than one, in some astonishment.

So, for example:

Homework: Reflect Triangle LMN [L: (-1,4), M: (0,7), N: (-4, 10) over line y=x+2. Prove it.

So we discussed the steps before they left. I actually posed it as a couple of questions.

  1. If you sketched this and just estimated points the reflection, what would be the key information you’d need to pin down to go from “estimation” to “actual answer”?
  2. Can you think of any coordinate geometry algorithms that might help you find these points?

And working with me, they came up with this procedure for each point:

  1. Find the equation of the line perpendicular to the reflection line.
  2. Find the solution to the reflection line and the perpendicular line. This solution is also the midpoint between the original point and its reflection.
  3. Using the original point and the midpoint, find the reflection point.
  4. Prove the reflection is accurate by establishing that the sides of the original triangle and the reflection are congruent.

And here it is, mapped out in Desmos—but honestly, it was much easier to do on graph paper. I just wanted to increase my own Desmos capability.


This is the cleaned up version. Maybe I should put the actual work product here. But I’m not very neat. Next time I’ll take pictures of some of the kids’ work; it’s gorgeous.

When we came in the next day, the kids excitedly told me they’d not only done the work, but “figured out how to do it without the work!” Sure, I said, and we then predicted what would happen with the reflection of y=x+3, y=-x + 4, and so on.

But what about reflecting it over the line y=-2x?


Gleesh. I didn’t have time during summer to investigate why the numbers are so ugly. The kids got tired after doing two points, and I told them to use calculators. But we did get it to work. We could see the fractions begin in the perpendicular line solutions, since we’re always adding .5x to 2x. But would it always be like that?

However, I’ve got one great activity for strong kids done–it reinforces knowledge of reflection, coordinate geometry, systems of equations, and some fairly messy algebra. Whoo and hoo.

Down side–for the first time in two years, I’m not teaching geometry this year!

All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.

Hey, under 1000!

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16 responses to “Isometries and Coordinate Geometry

  • Jim

    Regarding your ugly numbers – If l is the line in the plane with equation a.x = a.p with a0 the formula for reflection in the line l is

    r(x) = x – 2(x.(a-p)/a.a) a

    In your first example a=(1,-1) so a.a = 2 which is canceled out by the 2 in the formula so if x has integral coordinates and p has integral coordinates then r(x) has integral coordinates. Reflecting in the line y=-2x corresponds to a=(2,-1) and a.a=5 which is prime to 2 so no cancellation and you get answers with denominators of 5.

  • JIm

    Alternatively lines of slope +/-1 which go through a point with integral coordinates are lines of symmetry for the standard lattice consisting of points with integral coordinates but of course most lines are not lines of symmetry of this lattice.

  • Florida resident

    Dear Ed:
    I understand that the ugly (IMHO) word “congruent” has the meaning assigned to it by you and by the rest of mathematical community.

    However, it is worth teaching kids, that two triangles drawn by you
    can not be overlapped with each other by translations and/or or rotations _within_ the plane; to overlap them, you need to go out of plane (what you call reflections in the straight line.)

    In other words, one triangle is “right”, the other is “left”.

    Isosceles triangles are neutral with respect to right/ left dichotomy, and for them it is no problem.

    Best to you and to Gurbanguly Berdimuhamedow.

  • Jim

    Yes, of course, there is an important distinction being two figures which are “properly congruent” ie related by an orientation preserving isometry and the case when two figures are congruent only by an orientation-reversing isometry. This matter of orientation is pervasive throughout all geometry over the real field and can be traced back to the fact that zero disconnects the real field. Zero does not disconnect the complex field so in complex geometry the whole issue of orientation dissapears. Related to this feature is that a complex line in say a 2 dimensional complex space does not disconnect it. You can go around the complex line. Similarilly in complex geometry you never have something like say a 2-dimensional sphere which disconnects real 3-space. A connected complex manifold cannot be disconnected by a lower dimensional complex submanifold.

    Also in physics the role that orientation plays in the basic structure of the universe is a fascinating topic.

  • Jim

    The difference between zero disconnecting the real field b\ut not disconnecting the complex field is one of the single most important facts in all of mathematics. Shout it from the rooftops.

    • Florida resident

      Dear Jim:
      Thank you for your comment.
      To solve equation x+5=3,
      one had to introduce new numbers: negative ones.
      To solve equation x*7=6,
      one had to introduce new numbers: fractions, or rational numbers.
      To solve equation x^2=2/3,
      one had to introduce new numbers: irrational ones.
      Skip real transcendental numbers, like e or pi.
      To solve equation x^2 +1 = 0,
      one had to introduce new numbers: complex ones.

      However, to solve equation x^2 – i = 0,
      one _did_not_have_ to introduce new numbers like “hypercomplex”:
      x=(1+i)/sqrt(2) does the job.

      Any polynomial has a root within the field of complex numbers
      (Fundamental Theorem of Algebra).
      Shout it from the rooftops !

      With invariable respect of educationrealist,
      your F.r.

  • Jim

    Yes, the Fundamental Theorem of Algebra is also one of the great miracles of mathematics.

  • Anon

    “All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.”

    You’re doing matrix multiplication rotations through another method.

    r00 = Cosine(x) r01 = Sine(x)
    r10 = -1 * Sine(x) r11 = Cosine(x)

    NewX = x * r00 + y * r10;
    NewY = x * r01 + y * r11;

    In order to do the first one, you need to subtract 2 from the y coordinate, rotate by 315, flip the y sign,
    rotate by 45, and add 2.

    For the second one you need to rotate by 247.5 degrees, flip y sign, and rotate by 112.5.

    Just working one of them out:
    -4, 10

    -2 for y:

    -4, 8

    Sin(315) ~= -.707 or negative (square root of 2 / 2)
    Cos(315) ~= .707

    -4,8 =

    -4*(Sqrt(2)/2) + 8*(Sqrt(2)/2) = 4*(Sqrt(2)/2)
    4*(Sqrt(2)/2) + 8*(Sqrt(2)/2) = 12*(Sqrt(2)/2)

    Or: 2.828427125,8.485281375




    4*(Sqrt(2)/2), -12*(Sqrt(2)/2)

    Sin(45) ~= .707 or again (Sqrt(2)/2)
    Cos(45) ~= .707

    4*(Sqrt(2)/2) * (Sqrt(2)/2) + -12*(Sqrt(2)/2)* -1*(Sqrt(2)/2)
    = 4*(2/4) + (12*(2/4))
    = 2 + 6
    = 8

    4*(Sqrt(2)/2) * (Sqrt(2)/2) + -12*(Sqrt(2)/2)* (Sqrt(2)/2)
    = 4*(2/4) + (-12*(2/4))
    = 2 – 6
    = -4



    = 8,-2

    So this one was nice because you’re multiplying the square root of 2 with itself and some other things.

    The rotation for the second one is a great deal messier. the angle is 90+22.5 = 112.5

    Cos(247.5) = -.382683432
    Cos(112.5) = -.382683432

    Sin(247.5) = -.923879533
    Sin(112.5) = .923879533

    not doing the math for this one.

    This by contrast is apparently the square root of 0.146446609 and the square root of 0.853553391 and they aren’t canceling out.

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  • surfer

    What course covers reflection of a curve across a line? Is it part of 9th grade algebra? 10th grade geometry? 11th grade alg2/trig? 12th grade analytic geometry/precalc? Some other course further on?

    Just wonder where we do this? Worry a little that you are sort of inventing analytic geometry on your own, ad hoc, and the kids would get this later. Or in any case, there are texts that cover this type of transformation in an organized manner, so we could look at these for models/insights. But I admit I don’t remember when this is covered.

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