Question
If AB = 25 inches then find the distance of A'B' after a reflection over the x axis and a rotation of 17 degrees clockwise
Answers
GPT 3.5
To find the distance of A'B' after reflection over the x-axis and rotation of 17 degrees clockwise, we need to apply each transformation step by step.
1. Reflection over the x-axis:
When a point is reflected over the x-axis, the x-coordinate remains the same, and the y-coordinate is negated.
If A(x, y) is the original point, after reflection, it becomes A'(x, -y).
In this case, since A is not given, we cannot calculate its y-coordinate directly. However, we know that AB = 25 inches, which means that A and B lie on a line parallel to the x-axis. Therefore, we can assume that the y-coordinate of B is 0. So, B(25, 0) is the original point.
After reflection over the x-axis, B'(25, -0) = B'(25, 0) remains the same.
2. Rotation of 17 degrees clockwise:
To rotate a point (x, y) counterclockwise by a given angle, we use the rotation matrix formula:
[x'] [cosθ -sinθ] [x]
[y'] = [sinθ cosθ] [y]
where (x', y') are the coordinates of the new point, (x, y) are the coordinates of the original point, and θ is the angle of rotation.
In this case, (x, y) = B'(25, 0), and the angle of rotation θ = -17 degrees since we are rotating clockwise.
Applying the rotation matrix formula, we get:
[x'] [cos(-17) -sin(-17)] [25] [1 0.29237] [25] [25 * 1 + 0 * 25]
[y'] = [sin(-17) cos(-17)] [0] = [0.29237 0.95630] [0] = [25 * 0.29237 + 0 * 0.95630]
Simplifying the calculations:
[x'] [25]
[y'] = [7.30925]
Therefore, A'(x', y') = A'(25, 7.30925).
The distance of A'B' can be calculated using the distance formula:
distance = √((x' - x)^2 + (y' - y)^2)
= √((25 - 25)^2 + (7.30925 - 0)^2)
= √(0 + 53.4086760625)
≈ √53.4086760625
≈ 7.310 inches.
So, the distance of A'B' after the reflection over the x-axis and rotation of 17 degrees clockwise is approximately 7.310 inches.
1. Reflection over the x-axis:
When a point is reflected over the x-axis, the x-coordinate remains the same, and the y-coordinate is negated.
If A(x, y) is the original point, after reflection, it becomes A'(x, -y).
In this case, since A is not given, we cannot calculate its y-coordinate directly. However, we know that AB = 25 inches, which means that A and B lie on a line parallel to the x-axis. Therefore, we can assume that the y-coordinate of B is 0. So, B(25, 0) is the original point.
After reflection over the x-axis, B'(25, -0) = B'(25, 0) remains the same.
2. Rotation of 17 degrees clockwise:
To rotate a point (x, y) counterclockwise by a given angle, we use the rotation matrix formula:
[x'] [cosθ -sinθ] [x]
[y'] = [sinθ cosθ] [y]
where (x', y') are the coordinates of the new point, (x, y) are the coordinates of the original point, and θ is the angle of rotation.
In this case, (x, y) = B'(25, 0), and the angle of rotation θ = -17 degrees since we are rotating clockwise.
Applying the rotation matrix formula, we get:
[x'] [cos(-17) -sin(-17)] [25] [1 0.29237] [25] [25 * 1 + 0 * 25]
[y'] = [sin(-17) cos(-17)] [0] = [0.29237 0.95630] [0] = [25 * 0.29237 + 0 * 0.95630]
Simplifying the calculations:
[x'] [25]
[y'] = [7.30925]
Therefore, A'(x', y') = A'(25, 7.30925).
The distance of A'B' can be calculated using the distance formula:
distance = √((x' - x)^2 + (y' - y)^2)
= √((25 - 25)^2 + (7.30925 - 0)^2)
= √(0 + 53.4086760625)
≈ √53.4086760625
≈ 7.310 inches.
So, the distance of A'B' after the reflection over the x-axis and rotation of 17 degrees clockwise is approximately 7.310 inches.