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Common Core Standards: Math

High School: Geometry

Congruence HSG-CO.A.5

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Rotations, reflections, and translations might sound like moves in a Cirque du Soleil show. Rotating about the trapeze, reflected in a shiny disco ball, and translated from French to English. But really, rotations, reflections, and translations are all about staying rigid rather than flexible. Sorry, contortionists.

Students should already be well aware of rigid transformations, only now, they'll actually be using them instead of theorizing about them. Students should come up with sets of transformations (meaning more than one) that map figures onto themselves and other figures.

So, while not death-defying like some circus acts, what's exciting about this standard for students is that they are taking definitions and rules they've learned and actually bringing them into practice on paper.

Following this lesson, students should be able to use visuals to improve their sense and knowledge of how a transformation maps one shape onto another. They should also understand why moving and manipulating figures is a useful tool in real-world situations.

Make sure to keep your students motivated. This standard is the last of those dealing with working on transformations in the plane. Since they've already learned so much about congruence and transformations (or they should have, anyway), it's not as if they have to jump through hoops to master this concept.

All they have to do is put pencil to paper and start drawing. Well, transforming drawings, actually.

Drills

  1. Point A has the coordinates (-4, 4). If we want to reflect A across the y-axis to make a new point, B, what will the coordinates of B be?

    Correct Answer:

    (4, 4)

    Answer Explanation:

    Points A and B are direct mirror images of each other across the y-axis, so we have to find out which coordinates B will have. Visually, we can imagine folding the graph along the y-axis, ending up with a point that's 4 units away from the y-axis, but on the other side. The y coordinate stays the same, but the x coordinate changes from negative to positive. That means (B) is our answer.


  2. Which transformation is most likely to turn a square into a diamond?

    Correct Answer:

    Rotation

    Answer Explanation:

    Imagine a square. Now tilt your head about 45° to the right, and presto! It's a diamond. All we had to do was rotate it a little bit. Translation keeps the square as is, and reflection wouldn't be able to turn the square in the way we need. Since those two won't work, we're guessing that (D) probably won't work either.


  3. A triangle has vertices at A (2, 1), B (4, 4), and C (4, 1). Another triangle has coordinates at D (7, 3), E (9, 6), and F (9, 3). How many units must ΔABC be translated to carry onto ΔDEF?

    Correct Answer:

    5 units to the right, 2 units up

    Answer Explanation:

    If you were to line up your x and y coordinates for each vertex of the two triangles then plot them on a coordinate plane, you would see that the difference between all x coordinates is 5 and all y coordinates is 2. So to translate ΔABC onto ΔDEF, we need to move 5 units to the right, and 2 units up. All the coordinates should match up.


  4. On the x-y plane, ΔABC has coordinates of A (0, 5), B (0, 4), and C (4, 5), while ΔDEF has coordinates of D (-4, 0), E (-4, -1), and F (0, 0). How many units must ΔABC be translated to fit onto ΔDEF?

    Correct Answer:

    5 units down, 4 units to the left

    Answer Explanation:

    All it takes is finding the common differences between the coordinates. As always, we subtract the final from the initial. D (-4, 0) minus A (0, 5) gives us 4 units to the left and 5 units down. If we do the same thing for each pair of units (E minus B, and F minus C), we'll end up with 4 units to the left and 5 units down. That's conclusively (D).


  5. Which two transformation or transformations will create trapezoid A'B'C'D' from trapezoid ABCD?

    Correct Answer:

    Translation and reflection

    Answer Explanation:

    Since the horizontal sides of the trapezoid are switched, it should be obvious that translation isn't the only thing going on here. If we keep tabs on the points of the trapezoid, rotating it 180° would have A' switched with B' and C' switched with D'. Reflection across the x-axis, however, puts the points in the right place. Then, we have to translate the trapezoid over because an additional reflection across the y-axis would switch the points again. That means we need to perform translation and reflection.


  6. The Taj Mahal in India has a very obvious axis of symmetry. If we wanted to create an upside down Taj Mahal (for some reason), which of the following transformations could we use?

    Correct Answer:

    Rotation and reflection

    Answer Explanation:

    Translation keeps the figure exactly as is without changing its orientation. If we want it upside down, that changes its orientation. This effectively eliminates (A) and (C). Now, the only question is whether rotation can do the trick. Since the two sides of the Taj Mahal are symmetrical, whether the right side is on the left and the left on the right doesn't really matter. It'll still be an upside down version of the original one. Both rotation and reflection will work just fine.


  7. Which of the following is true about reflection?

    Correct Answer:

    Reflecting an image twice can be the same as rotating the image.

    Answer Explanation:

    Reflection, rotation, and translation are all different. Sometimes, multiple transformations can get us to the same place. If we reflect a shape twice, we may just end up putting it right back where it started, which isn't much of a rotation. However, reflecting a shape in a quadrant across the x and then y-axes, that would be the same as rotating the image. One reflection won't do it, but 2 reflections will.


  8. How many lines of symmetry does a five-pointed star have?

    Correct Answer:

    5

    Answer Explanation:

    If we draw an imaginary line through the top point of the star straight down, we can fold it in half perfectly. That makes one line of symmetry. Since all five points of the star are identical, we can apply this to every single point in the star. That means a total of 5 lines of symmetry.


  9. The points of rectangle are L (-4, 6), M (-1, 6), N (-1, 2), and O (-4, 2). The rectangle is first reflected across the y-axis and then translated down 4 units and to the left 1 unit. Which of the following are the correct coordinates of rectangle L'M'N'O'?

    Correct Answer:

    L' (3, 2), M' (0, 2), N' (0, -2), O' (3, -2)

    Answer Explanation:

    How does reflection affect the coordinates of a point (or 4 points)? If we're reflecting across the y-axis, we change the sign of all our x coordinates and keep the y's as they are. Then, we translate 4 down and 1 to the left. In other words, subtract the x coordinate by 1 and the y coordinate by 4. We only need to do this for L and M, and we should get (B) as our answer.


  10. The points of rectangle are L (-4, 6), M (-1, 6), N (-1, 2), and O (-4, 2). The rectangle is first translated down 4 units and to the left 1 unit and then reflected across the y-axis. Are the coordinates of the new rectangle the same as the coordinates of L'M'N'O' in the previous question?

    Correct Answer:

    No, because the transformations are performed in a different order

    Answer Explanation:

    The same set of transformations performed in a different order may change the resulting image. For instance, translating point (1, 1) upward 100 units and then reflecting it across the x-axis would make (1, -101). Performing the same transformations in the opposite order would give us (1, 99). Rigid transformations may still move images to different places and in different ways. That makes (C) the right answer.