AMNO with M(-2, 2), N(0, -2), O(1, 0); 180°. Where Is N?
To rotate point N(0, -2) 180° about point M(-2, 2), we can use the formula for a 2D rotation:
(x', y') = (x - a, y - b) rotated by θ
where (x, y) are the coordinates of the original point, (x', y') are the coordinates of the rotated point, (a, b) are the coordinates of the center of rotation, and θ is the angle of rotation.
Substituting the values into the formula, we have:
(x', y') = (0 - (-2), -2 - 2) rotated by 180°
Simplifying this, we get:
(x', y') = (2, -4) rotated by 180°
Therefore, the coordinates of the rotated point N are (2, -4).
To find the location of N after rotating the triangle AMNO by 180°, you can use the rotation formula.
The general formula for rotating a point (x, y) by an angle θ counterclockwise about the origin is:
x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)
In this case, the angle of rotation is 180°, so we have:
θ = 180°
Let's plug in the coordinates of N (0, -2) into the rotation formula:
x' = 0 * cos(180°) - (-2) * sin(180°)
y' = 0 * sin(180°) + (-2) * cos(180°)
Simplifying these equations gives us:
x' = 0 - 0 = 0
y' = 0 + 2 = 2
Therefore, after rotating the triangle AMNO by 180°, the new location of point N is (0, 2).
To find the position of point N after a 180° rotation of triangle AMNO, we need to rotate point N by 180° about a certain center of rotation.
To perform a 180° rotation, we can use the following steps:
1. Find the center of rotation: The center of rotation is the midpoint of the line segment that connects points A and O. In this case, point A is M(-2, 2) and point O is O(1, 0). The midpoint formula is given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Applying the formula, the midpoint of line segment AO is:
Midpoint = ((-2 + 1) / 2, (2 + 0) / 2)
= (-1/2, 1)
So, the center of rotation is C(-1/2, 1).
2. Determine the distance between the center of rotation and point N. The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Applying the formula, the distance between the center of rotation C(-1/2, 1) and point N(0, -2) is:
Distance = sqrt((0 - (-1/2))^2 + (-2 - 1)^2)
= sqrt((1/2)^2 + (-3)^2)
= sqrt(1/4 + 9)
= sqrt(1/4 + 36/4)
= sqrt(37/4)
= sqrt(37)/2
3. Find the direction (angle) of rotation. In this case, we need to perform a 180° rotation, which means the direction of rotation is clockwise.
4. Rotate point N by 180° about the center of rotation C. To do this, we need to find the new coordinates of N. The new x-coordinate is the reflection of the original x-coordinate across the center of rotation, and the new y-coordinate is the reflection of the original y-coordinate across the center of rotation.
Using the reflection formulas, the new coordinates of N after a 180° rotation about C(-1/2, 1) are:
New N = (2 * x1 - x2, 2 * y1 - y2)
= (2 * (-1/2) - 0, 2 * 1 - (-2))
= (-1 - 0, 2 + 2)
= (-1, 4)
Therefore, after a 180° rotation of triangle AMNO, the new position of point N will be N'(-1, 4).