7. If 1 || m and n || q, identify any of the following angles congruent to 6
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Answer:
Here's what I get.
Step-by-step explanation:
1. Rotate 180° about origin
The formula for rotation of a point (x,y) by an angle θ about the origin is
x' = xcosθ - ysinθ
y' = ycosθ + xsinθ
If θ = 180°, sinθ = 0 and cosθ = -1, and the formula becomes
x' = -x
y' = -y
The rule is then (x, y) ⟶ (-x, -y).
H: (-3, -5) ⟶ (3, 5)
J: (-5, -3) ⟶ (5, 3)
Q: (0, -1) ⟶ (0, 1)
The vertices of H'J'Q' are (3, 5), (5, 3), and (0, 1).
2. Rotation 90° counterclockwise about origin
cos90° = 0 and sin90° = 1
x' = xcos90° - ysin90° = -y
y' = ycos90° + xsin90° = x
The rule is then (x, y) ⟶ (-y, x).
B: (4, 5) ⟶ (-5, 4)
L: (5, 0) ⟶ (0, 5)
S: (2, 2) ⟶ (-2, 2)
The vertices of B'L'S' are (-5, 4), (0, 5), and (-2, 2).
3. Rotation 90° clockwise about origin
cos(-90°) = 0 and sin(-90°) = -1
x' = xcos(-90°) - ysin(-90°) = y
y' = ycos(-90°) + xsin(-90°) = -x
The rule is then (x, y) ⟶ (y, -x).
F: (1, -5) ⟶ (-5, -1)
H: (-2, -3) ⟶ (-3, 2)
U: (-4, -5) ⟶ (-5, 4)
The vertices of F'H'U' are (-5, -1), (-3, -2), and (-5, 4).
4. Rotate 180° about origin
The rule is (x, y) ⟶ (-x, -y).
J: (1, -1) ⟶ (-1, 1)
V: (2, 0) ⟶ (-2, 0)
Y: (5, -3) ⟶ (-5, 3)
R: (4, -3) ⟶ (-4, 3)
The vertices of J'V'Y'R' are (-1, 1), (-2, 0), (-5, 3), and (-4, 3).
The figures below show your shapes before and after the rotations.
