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Geometry

Transformations

Lesson

Four ways to move or resize a figure in the coordinate plane.

Translation

(x,y)→(x+a,y+b)(x, y) \to (x + a, y + b)

Slides every point by the same offset.

Reflection

  • Over x-axis: (x,y)→(x,−y)(x, y) \to (x, -y)
  • Over y-axis: (x,y)→(−x,y)(x, y) \to (-x, y)
  • Over y=xy = x: (x,y)→(y,x)(x, y) \to (y, x)

Rotation about origin

  • 90° CCW: (x,y)→(−y,x)(x, y) \to (-y, x)
  • 180°: (x,y)→(−x,−y)(x, y) \to (-x, -y)
  • 270° CCW (= 90° CW): (x,y)→(y,−x)(x, y) \to (y, -x)

Dilation centered at origin

(x,y)→(kx,ky)(x, y) \to (k x, k y)

∣k∣>1|k| > 1: enlarge. ∣k∣<1|k| < 1: shrink.

How to type your answer

Enter the new point as x,y. Example: 3,-4.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Translate (2,3) by (1,4)\text{Translate } (2, 3) \text{ by } (1, 4)

Problem 2

Reflect (4,5) over x-axis\text{Reflect } (4, 5) \text{ over x-axis}

Problem 3

Reflect (4,5) over y-axis\text{Reflect } (4, 5) \text{ over y-axis}

Problem 4

Rotate (3,0) 90° CCW\text{Rotate } (3, 0) \text{ 90° CCW}

Practice

Standard problems matching the lesson.

Problem 5

Translate (1,2) by (3,−1)\text{Translate } (1, 2) \text{ by } (3, -1)

Problem 6

Translate (−5,−3) by (5,5)\text{Translate } (-5, -3) \text{ by } (5, 5)

Problem 7

Reflect (6,−2) over x-axis\text{Reflect } (6, -2) \text{ over x-axis}

Problem 8

Reflect (−3,7) over y-axis\text{Reflect } (-3, 7) \text{ over y-axis}

Problem 9

Reflect (4,5) over y=x\text{Reflect } (4, 5) \text{ over } y = x

Problem 10

Rotate (2,0) 180°\text{Rotate } (2, 0) \text{ 180°}

Problem 11

Rotate (0,5) 90° CCW\text{Rotate } (0, 5) \text{ 90° CCW}

Problem 12

Rotate (1,1) 90° CCW\text{Rotate } (1, 1) \text{ 90° CCW}

Problem 13

Rotate (3,4) 180°\text{Rotate } (3, 4) \text{ 180°}

Problem 14

Rotate (2,3) 270° CCW\text{Rotate } (2, 3) \text{ 270° CCW}

Problem 15

Dilate (4,6) by factor 2\text{Dilate } (4, 6) \text{ by factor 2}

Problem 16

Dilate (10,15) by factor 1/5\text{Dilate } (10, 15) \text{ by factor } 1/5

Problem 17

Translate corner (1,1) by (2,3)\text{Translate corner } (1, 1) \text{ by } (2, 3)

Problem 18

Reflect (5,−5) over x-axis\text{Reflect } (5, -5) \text{ over x-axis}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Reflect (−3,4) over y=x\text{Reflect } (-3, 4) \text{ over } y = x

Problem 20

Rotate (4,−2) 90° CCW\text{Rotate } (4, -2) \text{ 90° CCW}

Problem 21

Rotate (−1,3) 180°\text{Rotate } (-1, 3) \text{ 180°}

Problem 22

Dilate (5,−10) by 0.4\text{Dilate } (5, -10) \text{ by 0.4}

Problem 23

Translate (3,7) by (−5,−2)\text{Translate } (3, 7) \text{ by } (-5, -2)

Problem 24

Reflect (0,4) over x-axis\text{Reflect } (0, 4) \text{ over x-axis}

Problem 25

Rotate (5,5) 90° CW\text{Rotate } (5, 5) \text{ 90° CW}

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