Use the diagram below to find the coordinates of AKJV after a rotation of 90° counterclockwise about the point
(-1, 1)
To find the coordinates of a point after a rotation of 90° counterclockwise about a given point, we can use the following formula for rotation:
(x', y') = (x - h, y - k)
(cosθ, sinθ)
+ (h, k)
Where (x, y) are the original coordinates of the point, (x', y') are the coordinates after rotation, (h, k) are the coordinates of the center of rotation, and θ is the angle of rotation.
In this case, the center of rotation is (-1, 1) and the angle of rotation is 90°.
For point A with coordinates (-2, 3), we have:
(x', y') = (-2 - (-1), 3 - 1)
(cos90°, sin90°)
+ (-1, 1)
= (-2 + 1, 3 - 1)
(0, 1)
+ (-1, 1)
= (-1, 2)
(0, 1)
+ (-1, 1)
= (-1, 3)
Therefore, after a rotation of 90° counterclockwise about the point (-1, 1), the coordinates of AKJV become (-1, 3).
To find the coordinates of AKJV after a rotation of 90° counterclockwise about the point (-1, 1), follow these steps:
Step 1: Determine the distance between the point (-1, 1) and point A by subtracting the x-coordinates and the y-coordinates:
dx = xA - (-1) = xA + 1
dy = yA - 1
Step 2: Apply the rotation matrix formula to find the new coordinates:
x' = dx * cos(θ) - dy * sin(θ)
y' = dx * sin(θ) + dy * cos(θ)
Since we're rotating counterclockwise by 90°, θ = 90° or θ = π/2.
Step 3: Plug in the values and calculate:
x' = (xA + 1) * cos(π/2) - (yA - 1) * sin(π/2)
y' = (xA + 1) * sin(π/2) + (yA - 1) * cos(π/2)
Given that point A has coordinates (3, 4), plug in these values into the formulas:
x' = (3 + 1) * cos(π/2) - (4 - 1) * sin(π/2)
y' = (3 + 1) * sin(π/2) + (4 - 1) * cos(π/2)
Step 4: Simplify and solve the equations:
x' = 4 * cos(π/2) - 3 * sin(π/2)
y' = 4 * sin(π/2) + 3 * cos(π/2)
Using the trigonometric values of π/2:
x' = 4 * 0 - 3 * 1 = -3
y' = 4 * 1 + 3 * 0 = 4
Therefore, the coordinates of AKJV after the rotation are (-3, 4).
To find the coordinates of AKJV after a rotation of 90° counterclockwise about the point (-1, 1), we will follow these steps:
1. Locate the point A in the diagram, which appears to be at the coordinates (1, 1).
2. Translate the entire diagram so that the point of rotation, (-1, 1), coincides with the origin (0, 0). We do this by subtracting the coordinates of the point of rotation from each point in the diagram, including A. Therefore, the new coordinates for A become (1 - (-1), 1 - 1), which simplifies to (2, 0).
3. Perform a counterclockwise rotation of 90° about the origin on the translated diagram. This can be done by swapping the x and y coordinates of each point and changing the sign of the new x coordinate. Since we only need to find the coordinates of AKJV, we will focus on these points:
- A: The new coordinates of A after the rotation become (0, 2).
- K: The new coordinates of K after the rotation can be found by swapping the x and y coordinates and negating the new x coordinate: K(-2, 1).
- J: The new coordinates of J after the rotation become (0, -1).
- V: The new coordinates of V after the rotation become (2, 1).
Therefore, after a rotation of 90° counterclockwise about the point (-1, 1), the new coordinates of AKJV are as follows:
- A: (0, 2)
- K: (-2, 1)
- J: (0, -1)
- V: (2, 1)