Complete the equation, identifying both lines of reflection that flipped the given figure onto itself.A (-4,3), B(5,3), C(5,-2) D(-4,-2)

Answer 1

The given figure is a rectangle with vertices A(-4,3), B(5,3), C(5,-2), and D(-4,-2).

To find the lines of reflection that flip the figure onto itself, we can first find the midpoint of each side of the rectangle, and then connect the midpoints to find the lines of reflection. Let's calculate the midpoints:

Midpoint of AB:
x-coordinate: (x₁ + x₂) / 2 = (-4 + 5) / 2 = 1/2
y-coordinate: (y₁ + y₂) / 2 = (3 + 3) / 2 = 3

Midpoint of BC:
x-coordinate: (x₁ + x₂) / 2 = (5 + 5) / 2 = 5
y-coordinate: (y₁ + y₂) / 2 = (3 - 2) / 2 = 1/2

Midpoint of CD:
x-coordinate: (x₁ + x₂) / 2 = (5 - 4) / 2 = 1/2
y-coordinate: (y₁ + y₂) / 2 = (-2 - 2) / 2 = -2

Midpoint of DA:
x-coordinate: (x₁ + x₂) / 2 = (-4 + 5) / 2 = 1/2
y-coordinate: (y₁ + y₂) / 2 = (3 - 2) / 2 = 1/2

Now, let's connect the midpoints to determine the lines of reflection:

Line of reflection passing through the midpoints of AB and CD:
Connecting the points (1/2, 3) and (1/2, -2) gives us a vertical line passing through x = 1/2.

Line of reflection passing through the midpoints of BC and DA:
Connecting the points (5, 1/2) and (-4, 1/2) gives us a horizontal line passing through y = 1/2.

So, the two lines of reflection that flip the given figure onto itself are x = 1/2 (a vertical line passing through x = 1/2) and y = 1/2 (a horizontal line passing through y = 1/2).