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Reflected Triangles


Alignments to Content Standards: G-CO.A.5 G-CO.D.12

Task

The triangle in the upper left of the figure below has been reflected across a line into the triangle in the lower right of the figure. Use a straightedge and compass to construct the line across which the triangle was reflected.

Triangles_reflected__c13bdfa5e997fb8d05dc59b18088cd86

IM Commentary

This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.

Solution

The line in question is the perpendicular bisector of any pair of corresponding points. We choose the two corresponding points labelled $A$ and $B$ for the illustration below, but either of the other two pairs of points works just as well.

Solutionreflection_4c44343ec536642f457cdf70949d40fc

To summarize the construction: We construct the circle of radius $\overline{AB}$ cenetered at $A$ and the circle of radius $\overline{AB}$ cenetered at $B.$ These intersect at two points, which we label $P$ and $Q$. Then $P$ and $Q$ are both equidistant from $A$ and $B$, and so lie on the perpendicular bisector of $\overline{AB}$. We conclude that line $PQ$ is precisely the perpendicular bisector, which is the line over which the first triangle was reflected to arrive at the second.

We remark that the above construction is valid even from the strict Euclidean perspective on straight-edge and compass constructions. In a more modern setting, in which one typically allows the use of a compass with memory, one has a little more flexibility. In particular, we could replace the two circles in the above construction with any pair of circles of equal radius at least $\frac{1}{2}\overline{AB}$, centered at $A$ and $B$.

Jennifer says:

about 3 years

I blogged about using this task with my students: http://easingthehurrysyndrome.wordpress.com/2013/09/02/reflected-triangles/

stickler says:

about 5 years

This task should not be aligned to G.CO.5, because it neither asks students to draw the image of a figure under a transformation, nor does it ask students to describe a sequence of (i.e. more than one) transformations that maps a figure onto another. The wording of G.CO.5 is fairly straightforward, and this activity does not assess it. It would be better to consider this task to be aligned to G.CO.12, because it uses construction tools and the method of finding a perpendicular bisector of a segment (the segment formed by a point and its distinct image). Constructing a perpendicular bisector is specifically mentioned in the text of G.CO.12.

In order to adjust this task to really fit G.CO.5, one approach would be to give an original triangle and a line of reflection, then have students draw the image triangle. One typical approach would be to use a compass setting larger than the distance from the point to the line, and retaining that setting, create a rhombus to find an image point. This approach could be repeated for each vertex of the original figure to find the image points of the image figure.

Cam says:

about 5 years

Dear stickler,

Thanks for the careful reading of this task -- it spurned a lengthy discussion behind the scenes about the alignment. I agree that this task illustrates G-CO.12, and have added that alignment. I disagree that it does not illustrate G-CO.5, as part of that standard is to specify which transformations carry a given figure to another, which I think this task certainly does involve. (There is a semantic quibble about whether a single transformation can constitute a sequence, but I think both linguistically and pedagogically we are safe in saying that it does.)

Monique Rousselle Maynard says:

about 5 years

I look at this task and like its "backward" approach as an alternative to assessing if students truly grasp the concept of reflections. I anticipate students might first reach for a ruler, measure the lengths and determine midpoints between corresponding points of pre-image and image, and use these to sketch a line of reflection. Additionally, I see technology like a ClassPad at play here or inclusion of the coordinate plane where students are asked to incorporate algebra and determine the equation for the line of reflection.

Kristin says:

about 5 years

Monique, this is a great addition to the commentary provided in the task. Thanks!