# Common Core Standards: Math

### Geometry 8.G.A.3

3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Transformations are fascinating things in real life, in movies, and of course, in geometry. But if your students expect shapes to turn into giant fighting robots, they've got another thing coming.

Students should already be aware of the basic geometric transformations and understand how they change and move shapes on the plane. This can be as simple as knowing that translations are slides, rotations are turns, reflections are flips, and dilations are enlargements or contractions.

Obviously, that isn't good enough. Did your students really think that words could describe mathematical concepts better than numbers? Please, don't make us laugh.

When shapes are on the coordinate plane, we can describe the transformations they undergo using their coordinates. Translations can be described by adding or subtracting numbers to the x and y coordinates of a shape for horizontal and vertical movement, respectively. Dilations from the origin involve multiplying each of an object's coordinates by the same number. Otherwise, drawing the figures out on the x-y plane and measuring distances might be the best way to go.

Rotations and reflections might be a bit trickier, but we can still use coordinates to describe these transformations. We often rotate shapes around a central point, which means our image should be the same distance away from the central point as the preimage was. With reflections, we have to know which line is our axis of symmetry. Often, this involves flipping negative signs.

Once students know how to use coordinates this way, they should be able to convert between words, numbers, and images and express transformations. We won't expect them to use transformations to create enormous battling machines, but you might want to offer a little extra credit—or perhaps a movie deal—to those who do.

#### Drills

1. Point A, with coordinates (4, 6) is translated four units to the right. What are the coordinates of A'?

(8, 6)

Translation is fancy talk for sliding. To represent translation using coordinates, we add or subtract numbers to the appropriate coordinate. Moving four units to the right means a horizontal shift, so we'll be adding 4 to the x coordinate. That should give us (D) as our answer because 4 + 4 = 8, and since there's no vertical movement, the y coordinate stays the same. All the other answers either subtract 4 instead of adding it or add it to the y coordinate instead.

2. If point A, with coordinates (0, 2) is translated to point A', with coordinates (1, -1), which translation did A undergo?

One unit right and three units down

If we find the difference between the coordinates A' and A, we'll find out how point A was translated. Subtracting the x values gives 1 – 0 = 1 and subtracting the y values gives -1 – 2 = -3. Since x coordinates correspond to horizontal movement and y coordinates correspond to vertical movement, point A was moved to the right one unit and down three units. All other answers are incorrect either in the direction of the movement or the number of units moved.

3. A circle has a center at (-4, 3) and a radius of 1. If it's rotated 180° around the origin, which of the following will be the new center of the circle?

(4, -3)

It doesn't matter whether the circle is rotated clockwise or counterclockwise around the origin because it'll end up in the same place. The center of the circle should be in the quadrant opposite that of the original circle. Since the center is originally in quadrant II, the new center will be in quadrant IV. This eliminates (A) and (D). Since the rotation is 180°, we should be able to draw a straight line through the old center, the origin, and the new center. If we do this correctly, we should end up with (B) as the answer.

4. A figure has been transformed by moving a figure 4 units to the right and 2 units up. Which rule will transform it back to where it was?

(x, y) → (x – 4, y – 2)

To "undo" moving 4 units to the right and 2 units up, you move 4 units to the left and 2 units down. This means subtracting 4 from the horizontal (x) coordinate and 2 from the vertical (y) coordinate. While (B) and (D) look eerily similar, remember that order matters when it comes to subtraction. They aren't the same thing.

5. A square with coordinate (1, 1), (2, 1), (1, 2), and (2, 2) is dilated by 3 from the origin. Which of the following is one of the coordinates of the new square?

(6, 3)

Dilation is a tricky transformation, but since the center of dilation is the origin, all we need to do is multiply the coordinates by the scale of dilation. In this case, multiplying each coordinate by 3 would give us (3, 3), (6, 3), (3, 6), and (6, 6). Of the answer choices, only (A) is one of these four points. The rest are just plain wrong.

6. ΔABC has the points A (4, 3), B (9, 4), and C (7, 9). If it's reflected across the line x = 1, which of the following is a point on the new triangle?

(-7, 4)

Nothing can replace the utility of actually plotting out these points on a grid. Once we do that and realize that x = 1 is vertical (not horizontal!), we can find the distances between the points and the line of symmetry. The y coordinates will stay the same, but the x coordinates won't. Since (9, 4) is 9 – 1 = 8 units away from x = 1, we'll have to move it 8 units over, which gives us an x value of 1 – 8 = -7. The right answer is (C), and all the others come from miscalculating or accidentally assuming the axis of symmetry was at y = 1.

7. A circle with a radius of 4 is dilated by a factor of 3. The center of dilation is the same point as its center, (2, 5). Which of the following is a point on the new circle?

(-10, 5)

Our center of dilation is the center of the circle, which means we extend the new circle outward from there. In other words, (2, 5) will be the center for our new circle also—but don't choose (B) so quickly. We're looking for a point on the new circle, which has a radius that's three times as long as the old one. The radius for the new circle is 12, which means that any point that's exactly 12 units away from (2, 5) will be on the dilated circle. The only point that's 12 units away from (2, 5) is (D).

8. A point at coordinates (0, 5) is rotated 90° counterclockwise around the origin and then translated 3 units to the right and 2 units up. What are the coordinates of the new point?

(-2, 2)

A lot of stuff is happening here, so let's take it step by step. First, rotating the point (0, 5) around the origin 90° counterclockwise means it ends up at (-5, 0). So far, so good. After that, we move it 3 units to the right and 2 units up, or (x, y) → (x + 3, y + 2). Plugging in our point post-rotation gives us (-5 + 3, 0 + 2), or (-2, 2). The other answer options result from rotating the point clockwise or translating the coordinates incorrectly.

9. Which series of transformations would move a point at (3, 1) to (-3, 1)?

A rotation of 180° around the origin followed by a translation of 2 units up

Never underestimate the power of drawing. Not only can it be an emotional and creative outlet, but it can help you with geometry too. Answer (A) would be right if the reflection were across x = 3, not y = 3. For (C) to be right, it would need a rotation of 180°, not 90°, and that last translation bit makes (D) incorrect. Rotating (3, 1) by 180° would move it to (-3, -1) and a translation of 2 up would give us the (-3, 1) coordinates we want.

10. Which series of transformations would change a square with points (-2, 2), (-2, -2), (2, 2), and (2, -2) to a square with points (0, 0), (6, 6), (0, 6), and (6, 0)?

A translation of 2 units to the right and 2 units up followed by a dilation of 1.5 from the origin