Question

Rotate quadrilateral ABCD with vertices A(3,5), B(1,6), C(2,1), D(4,0) 90° ccw about (1,1).

a) A'(2,3)
b) A'(1,2)
c) A'(4,4)
d) A'(0,5)

Answer 1

Main Answer:

The rotation of point A(3,5) about (1,1) by 90° counterclockwise results in A'(1,2) in the new position. b) A'(1,2).

Therefore, the correct answer is b) A'(1,2).

Step-by-step explanation:

To rotate quadrilateral ABCD 90° counterclockwise about (1,1), the new coordinates of point A, denoted as A', can be found. The rotated position of A is (1,2).

In the rotation process, the key is to use the rotation formula for a point (x, y) about a center (h, k) by an angle θ:

`\[x' = (x - h) \cdot \cos(\theta) - (y - k) \cdot \sin(\theta) + h\]\[y' = (x - h) \cdot \sin(\theta) + (y - k) \cdot \cos(\theta) + k\]`

Here, A(3,5) is rotated 90° counterclockwise about (1,1). Plugging in the values:

`\[x' = (3 - 1) \cdot \cos(90°) - (5 - 1) \cdot \sin(90°) + 1 = 1\]\[y' = (3 - 1) \cdot \sin(90°) + (5 - 1) \cdot \cos(90°) + 1 = 2\]`

Thus, A' is located at (1,2) after the rotation.

Therefore, the correct answer is b) A'(1,2).