# Kite KLMN has vertices at K(1, 3), L(2,4), M(3,3), and N(2,0). After the kite is rotated, K' has coordinates (-3,1). How do you find the rotation, and include a rule in your description. Then find the coordinates of L', M',and N'?

Rotated about origin by

Given :

I.e. rotated

That means,

Hence, L((2),(4)) -> L’((-4),(2))#

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To find the rotation that was applied to the kite, you can use the coordinates of the original point K and its image K'. The rule for rotation about a point is:

[ (x', y') = (x_0 + (x - x_0) \cdot \cos(\theta) - (y - y_0) \cdot \sin(\theta), ] [ y_0 + (x - x_0) \cdot \sin(\theta) + (y - y_0) \cdot \cos(\theta)), ]

where ( (x_0, y_0) ) are the coordinates of the center of rotation, ( (x, y) ) are the coordinates of the point to be rotated, ( (x', y') ) are the coordinates of the image after rotation, and ( \theta ) is the angle of rotation.

Using the given coordinates of K(1, 3) and its image K'(-3, 1), you can substitute these values into the rotation formula to solve for ( \theta ).

After finding the angle of rotation, you can apply the same rotation to the remaining vertices of the kite to find their new coordinates. Here are the steps:

- Find the angle of rotation ( \theta ) using the coordinates of K and K'.
- Apply the same rotation to points L, M, and N to find their new coordinates.

Given K'(-3, 1), we have: [ x_0 = 1, \quad y_0 = 3, \quad x = -3, \quad y = 1. ] Substitute these into the rotation formula: [ -3 = 1 + (1 - 1) \cdot \cos(\theta) - (1 - 3) \cdot \sin(\theta) ] [ 1 = 3 + (1 - 1) \cdot \sin(\theta) + (1 - 3) \cdot \cos(\theta) ]

Solving these equations will give you the value of ( \theta ). After finding ( \theta ), you can use it to rotate the coordinates of L(2, 4), M(3, 3), and N(2, 0) to find their new coordinates L', M', and N'.

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