To reflect quadrilateral ABCD across the horizontal line through the midpoint of BC, find the midpoint, use its y-coordinate to define the line of reflection, then reflect each point across this line. The reflected quadrilateral has vertices at A'(-3,7), B'(8,8), C'(6,2), D'(0,10), and the line of reflection is y = 5.
To reflect quadrilateral ABCD in the horizontal line that passes through the midpoint of segment BC, we first need to find the midpoint of BC. The coordinates of B and C are B(8,8) and C(6,2), respectively.
The midpoint, M, of a segment with endpoints (x1, y1) and (x2, y2) is given by M = ((x1 + x2)/2, (y1 + y2)/2). Hence, the midpoint of BC is M = ((8 + 6)/2, (8 + 2)/2) = (7, 5).
This midpoint will be a point on the line of reflection. Since the line of reflection is horizontal and passes through M, its equation is y = 5. To find the image of each vertex after reflection, we take each y-coordinate and find its distance to the line y = 5, then move that vertex the same distance on the other side of the line.
Reflecting point A(-3,3): The distance from A to the line of reflection is |3 - 5| = 2. So, the image of A, which we'll call A', will be at (-3,5 + 2) = (-3,7).
Since points B and C are on the line of reflection, their images will be the same as their original positions. So, B' is at (8,8) and C' is at (6,2).
Reflecting point D(0,0): The distance from D to the line of reflection is |0 - 5| = 5. So, the image of D, which we'll call D', will be at (0,5 + 5) = (0,10).
The reflected quadrilateral A'B'C'D' has vertices at A'(-3,7), B'(8,8), C'(6,2), and D'(0,10). The line of reflection is y = 5, and this geometric transformation results in a symmetrical image across the horizontal line.
Below is the attached graph