1. Five-Minute Check (over Lesson 3–2) Mathematical Practices Then/Now
New Vocabulary
Key Concept: Rotation
Example 1: Draw a Rotation
Key Concept: Rotations in the Coordinate Plane
Example 2: Rotate a Figure About the Origin
Example 3: Rotate About a Point Other Than the Origin
Lesson Menu

2. Find the coordinates of the figure under the given translation
Find the coordinates of the figure under the given translation. RS with endpoints R(1, –3) and S(–3, 2) along the translation vector 2, –1
___
A. R'(–2, –2), S'(–1, 1)
B. R'(0, –3), S'(–5, 3)
C. R'(3, –4), S'(–1, 1)
D. R'(3, –4), S'(–5, 3)
5-Minute Check 1

3. Find the coordinates of the figure under the given translation
Find the coordinates of the figure under the given translation. ΔABC with vertices A(–4, 3), B(–2, 1), and C(0, 5) under the translation (x, y) → (x + 3, y – 4)
A. A'(–2, 1), B'(1, –3), C'(3, –1)
B. A'(–1, –1), B'(1, –3), C'(3, 1)
C. A'(0, 5), B'(–6, 3), C'(4, 7)
D. A'(1, –1), B'(2, 5), C'(5, 9)
5-Minute Check 2

4. Find the coordinates of the figure under the given translation
Find the coordinates of the figure under the given translation. trapezoid LMNO with vertices L(2, 1), M(5, 1), N(1, –5) and O(0, –2) under the translation (x, y) → (x – 1, y + 4)
A. L'(1, 5), M'(4, 5), N'(0, –1), O'(–1, 2)
B. L'(2, 6), M'(5, 7), N'(1, 0), O'(0, 3)
C. L'(3, –3), M'(6, –2), N'(0, –8), O'(–1, –6)
D. L'(4, –4), M'(7, 5), N'(0, –1), O'(1, 4)
5-Minute Check 3

5. Find the translation that moves AB with endpoints A(2, 4) and B(–1, –3) to A'B' with endpoints A'(5, 2) and B'(2, –5).
___
____
A. (x – 2, y – 3)
B. (x + 2, y + 2)
C. (x – 3, y + 2)
D. (x + 3, y – 2)
5-Minute Check 4

6. The preimage of rectangle ABCD has vertices at A(–4, 5), B(–4, –3), C(1, –3), and D(1, 5). Its image has vertices at A'(–1, 3), B'(–1, –5), C'(4, –5), and D'(4, 3). Write the ordered pair that describes the transformation of the rectangle.
A. (x, y) → (x + 3, y – 2)
B. (x, y) → (x – 3, y + 2)
C. (x, y) → (x + 2, y + 3)
D. (x, y) → (x – 2, y – 3)
5-Minute Check 5

7. Mathematical Practices 5 Use appropriate tools strategically.
7 Look for and make use of structure.
Content Standards
G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.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. MP

8. Given a geometric figure and a rotation, draw the transformed figure.
You identified rotations and verified them as congruence transformations.
Given a geometric figure and a rotation, draw the transformed figure.
Describe the effects of rotations on the coordinate plane.
Then/Now

9. center of rotation
angle of rotation
Vocabulary

10. Draw a segment from point R to point A.
Draw a Rotations
Copy quadrilateral RSTV and point A. Then use a protractor and ruler to draw a 45° rotation of RSTV about point A.
Draw a segment from point R to point A.
Use a protractor to measure a 45° angle counterclockwise with as one side. Extend the other side to be longer than AR.
Example 1

11. Locate point R' so that AR = AR'.
Draw a Rotations
Locate point R' so that AR = AR'.
Repeat this process for points S, T, and V.
Connect the four points to form R'S'T'V'.
Example 1

12. Draw a Rotations
Quadrilateral R'S'T'V' is the image of quadrilateral RSTV under a 45° counterclockwise rotation about point A.
Answer:
Example 1

13. For the diagram, which description best identifies the rotation of triangle ABC around point Q?
A. 20° clockwise
B. 20° counterclockwise
C. 90° clockwise
D. 90° counterclockwise
Example 1

14. Concept

15. First, draw ΔDEF and plot point G.
Rotate a Figure About the Origin
Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 115° clockwise about the point G(–4, –2).
First, draw ΔDEF and plot point G.
Draw a segment from point G to point D.
Use a protractor to measure a 115° angle clockwise with as one side.
Draw
Use a compass to copy onto Name the segment
Repeat with points E and F.
Example 2

16. Rotate a Figure About the Origin
Answer: ΔD'E'F' is the image of ΔDEF under a 115° clockwise rotation about point G.
Example 2

17. Triangle ABC has vertices A(1, –2), B(4, –6), and C(1, –6)
Triangle ABC has vertices A(1, –2), B(4, –6), and C(1, –6). Draw the image of ΔABC under a rotation of 70° counterclockwise about the point M(–1, –1).
A. B.
C. D.
Example 2

18. Rotate About a Point Other Than the Origin
Triangle JKL has vertices J(−2, 0), K(−1, −4), and L(−4, −3). Graph ΔJKL after a rotation 270° about the point (−2, 1).
Example 3

19. Step 2 Map the center of rotation to the origin.
Rotate About a Point Other Than the Origin
Step 1 Make a prediction.
Graph ∆JKL and predict where the image of ∆JKL will be on the coordinate plane after the given rotation.
Prediction: The image of ∆JKL after a 270° rotation about the point (–2, 1) will be a triangle in the second quadrant.
Step 2 Map the center of rotation to the origin.
To map the center of rotation to the origin, translate the center of rotation along the vector Then translate the vertices of ∆JKL along the same vector.
Example 3

20. Step 3 Rotate 270° about the origin.
Rotate About a Point Other Than the Origin
(x, y) → (x + 2, y – 1)
J(–2, 0) → (0, –1)
K(–1, –4) → (1, –5)
L(–4, –3) → (–2, –4)
Step 3 Rotate 270° about the origin.
(x, y) → (y, –x)
(0, –1) → (–1, 0)
(1, –5) → (–5, –1)
(–2, –4) → (–4, 2)
Example 3

21. Step 4 Map the center of rotation to its original location.
Rotate About a Point Other Than the Origin
Step 4 Map the center of rotation to its original location.
To map the center of rotation to its original location, translate the center of rotation along the vector Then translate the vertices of the rotated triangle along the same vector.
(x, y) → (x – 2, y + 1)
(–1, 0) → J′(–3, 1)
(–5, –1) → K′(–7, 0)
(–4, 2) → L′(–6, 3)
Example 3

22. Rotate About a Point Other Than the Origin
Answer:
Check The image matches the prediction because the image is in the second quadrant.
Example 3

23. Triangle PQR is shown below
Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin?
A. (–5, –4)
B. (–5, 4)
C. (5, 4)
D. (4, –5)
Example 3