Design your real-world scenario involving a geometric figure and its transformations. Model the pre-image and image on a coordinate plane. In your final answer, include the real-world scenario, written in complete sentences, the transformations that map the pre-image onto its image, and all calculations involved for the equations of the corresponding lines or the coordinates for the corresponding vertices. Also, please include a sketch of the pre-image and the image.
Real-World Scenario: A kite flying high in the sky undergoes a transformation when a strong gust of wind causes it to rotate and change its position.
Pre-Image: Let's consider the pre-image of the kite as a geometric figure represented by the coordinates (2, 4), (5, 7), (8, 4), and (5, 1). These points form the shape of a diamond, resembling a kite, on a coordinate plane.
image: The gust of wind causes the kite to rotate 45 degrees counterclockwise and move 2 units to the right and 3 units upward.
Transformations:
1. Rotation: To rotate the kite counterclockwise by 45 degrees, we can use the rotation formula:
(x', y') = (x * cos θ - y * sin θ, x * sin θ + y * cos θ)
Applying the formula to each vertex of the pre-image's coordinates, we get:
(2, 4) -> (2 * cos 45 - 4 * sin 45, 2 * sin 45 + 4 * cos 45) -> (√2, 6 - √2)
(5, 7) -> (5 * cos 45 - 7 * sin 45, 5 * sin 45 + 7 * cos 45) -> (-√2 + 6, 12)
(8, 4) -> (8 * cos 45 - 4 * sin 45, 8 * sin 45 + 4 * cos 45) -> (10 - √2, 6 + √2)
(5, 1) -> (5 * cos 45 - 1 * sin 45, 5 * sin 45 + 1 * cos 45) -> (3√2, 4)
2. Translation: To move the rotated kite 2 units to the right and 3 units upward, we add these values to the coordinates obtained from the rotation:
(√2, 6 - √2) -> (√2 + 2, 6 - √2 + 3) -> (√2 + 2, 9 - √2)
(-√2 + 6, 12) -> (-√2 + 6 + 2, 12 + 3) -> (-√2 + 8, 15)
(10 - √2, 6 + √2) -> (10 - √2 + 2, 6 + √2 + 3) -> (12 - √2, 9 + √2)
(3√2, 4) -> (3√2 + 2, 4 + 3) -> (3√2 + 2, 7)
Sketch:
The pre-image kite (diamond) vertices on a coordinate plane:
(2, 4) (5, 7)
/\ /\
/ \ / \
/ \ / \
(5, 1) (8, 4)
The image of the kite after rotation and translation:
(√2 + 2, 9 - √2) (10 - √2, 6 + √2)
/\ /\
/ \ / \
/ \ / \
(3√2 + 2, 7) (-√2 + 8, 15)
This real-world scenario depicts a kite being blown by the wind, rotating and moving in the sky. The transformations involved are a 45-degree counterclockwise rotation and a translation of 2 units to the right and 3 units upward.
OK.
Now it's your turn.