Consider a quadrilateral ABCD where A is at (1,3), B is at (4,3), C is at (6,8), and D is at (2,8). Next, the shape is reflected over the line y=2x. After that, a circle of radius 1 centered at point E is cut out of the quadrilateral. Point E is at (2,5). Finally, the shape undergoes a dilation centered at the origin with a scale factor of 2. What is the area of the resulting shape?
Question
Consider a quadrilateral ABCD where A is at (1,3), B is at (4,3), C is at (6,8), and D is at (2,8). Next, the shape is reflected over the line y=2x. After that, a circle of radius 1 centered at point E is cut out of the quadrilateral. Point E is at (2,5). Finally, the shape undergoes a dilation centered at the origin with a scale factor of 2. What is the area of the resulting shape?
Solution
This problem involves several steps, including reflection, circle area calculation, and dilation. Here's how to solve it:
Step 1: Reflecting the Quadrilateral Over the Line y=2x The reflection of a point (x, y) over the line y=2x is given by the point (x', y') where x' = (x+y)/3 and y' = (2x+y)/3. Applying this transformation to each point of the quadrilateral, we get the new coordinates:
A' = ((1+3)/3, (21+3)/3) = (4/3, 5/3) B' = ((4+3)/3, (24+3)/3) = (7/3, 11/3) C' = ((6+8)/3, (26+8)/3) = (14/3, 20/3) D' = ((2+8)/3, (22+8)/3) = (10/3, 12/3)
Step 2: Cutting Out the Circle The area of the circle centered at E with radius 1 is given by πr^2 = π1^2 = π. The area of the quadrilateral is not given, but since the circle is cut out from it, we subtract the area of the circle from the area of the quadrilateral.
Step 3: Dilation of the Shape A dilation of a shape from the origin by a scale factor of 2 doubles the area of the shape. So, the area of the resulting shape after dilation is 2*(Area of Quadrilateral - Area of Circle).
However, without the area of the quadrilateral, we cannot calculate the exact area of the resulting shape. We would need to calculate the area of the quadrilateral ABCD first, then subtract the area of the circle, and finally multiply by the scale factor of the dilation.
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