Let’s solve each problem step by step. We’re rotating shapes on the coordinate plane about the origin (0,0). Here are the rules we’ll use:
-
Rotation 90° clockwise: (x, y) → (y, -x)
-
Rotation 90° counterclockwise: (x, y) → (-y, x)
-
Rotation 180°: (x, y) → (-x, -y)
We’ll apply these to each point of the shape.
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Problem 1: Rotation 180° about the origin
Original points from graph (estimate from grid):
- O is at (0, 0) — stays at (0, 0)
- J is at (-4, -2) → becomes (4, 2)
- H is at (-3, -4) → becomes (3, 4)
So new triangle has vertices:
O'(0, 0), J'(4, 2), H'(3, 4)
Plot those and connect them.
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Problem 2: Rotation 90° counterclockwise about the origin
Original points:
- S is at (1, 1) → (-1, 1) [Wait: rule is (x,y)→(-y,x)] → so (1,1) → (-1, 1)? Let's check:
Actually: (x, y) → (-y, x)
So S(1,1) → (-1, 1) — correct? Wait: no!
Wait: (x=1, y=1) → (-y, x) = (-1, 1) — yes.
But let’s get exact coordinates from graph:
Looking at Problem 2 graph:
- L is at (3, 0) → rotate 90° CCW: (0, 3)
- S is at (1, 1) → (-1, 1)
- B is at (2, 3) → (-3, 2)
Check:
L(3,0) → (-0, 3) = (0,3)
✔
S(1,1) → (-1,1)
✔
B(2,3) → (-3,2)
✔
New triangle: L’(0,3), S’(-1,1), B’(-3,2)
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Problem 3: Rotation 90° clockwise about the origin
Rule: (x,y) → (y, -x)
Points from graph:
- F is at (0, 2) → (2, 0)
- B is at (-3, 1) → (1, 3)
- M is at (-3, -2) → (-2, 3)
- H is at (0, -2) → (-2, 0)
Wait — let’s list all four points of quadrilateral:
From graph:
F(0,2), B(-3,1), M(-3,-2), H(0,-2)
Apply rotation 90° clockwise: (x,y) → (y, -x)
- F(0,2) → (2, 0)
- B(-3,1) → (1, 3)
- M(-3,-2) → (-2, 3)
- H(0,-2) → (-2, 0)
New quadrilateral: F’(2,0), B’(1,3), M’(-2,3), H’(-2,0)
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Problem 4: Rotation 180° about the origin
Points:
U(-4, -3), H(-2, -1), V(0, -3)
Rotate 180°: (x,y) → (-x, -y)
- U(-4,-3) → (4,3)
- H(-2,-1) → (2,1)
- V(0,-3) → (0,3)
New triangle: U’(4,3), H’(2,1), V’(0,3)
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Problem 5: Rotation 90° clockwise about the origin
Given points:
U(1, -2), W(0, 2), K(3, 2), G(3, -3)
Rule: (x,y) → (y, -x)
- U(1,-2) → (-2, -1)
- W(0,2) → (2, 0)
- K(3,2) → (2, -3)
- G(3,-3) → (-3, -3)
New points: U’(-2,-1), W’(2,0), K’(2,-3), G’(-3,-3)
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Problem 6: Rotation 180° about the origin
Given points:
V(2, 0), S(1, 3), G(5, 0)
Rule: (x,y) → (-x, -y)
- V(2,0) → (-2, 0)
- S(1,3) → (-1, -3)
- G(5,0) → (-5, 0)
New points: V’(-2,0), S’(-1,-3), G’(-5,0)
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Now, since the student needs to *graph* the images, but we can’t draw here, we provide the transformed coordinates for each problem so they can plot them.
Final Answer:
1) After 180° rotation: O(0,0), J(4,2), H(3,4)
2) After 90° CCW rotation: L(0,3), S(-1,1), B(-3,2)
3) After 90° CW rotation: F(2,0), B(1,3), M(-2,3), H(-2,0)
4) After 180° rotation: U(4,3), H(2,1), V(0,3)
5) After 90° CW rotation: U(-2,-1), W(2,0), K(2,-3), G(-3,-3)
6) After 180° rotation: V(-2,0), S(-1,-3), G(-5,0)
Plot these new points on their respective grids and connect them in the same order as original figures.
Parent Tip: Review the logic above to help your child master the concept of rotations worksheet pdf.
