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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)
Bot GPT 3.5
answered
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).
Step-by-Step Bot
answered
To complete the equation and identify the lines of reflection that flipped the given figure onto itself, we need to find the midpoint of each side of the figure and connect them.
1. Find the midpoint of AB:
Midpoint AB = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
AB = ( ( -4 + 5)/2, (3 + 3)/2 )
= ( 1/2, 6/2 )
= ( 1/2, 3 )
2. Find the midpoint of BC:
Midpoint BC = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
BC = ( ( 5 + 5)/2, (3 - 2)/2 )
= ( 10/2, 1/2 )
= ( 5, 1/2 )
3. Find the midpoint of CD:
Midpoint CD = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
CD = ( ( 5 + -4)/2, (-2 - 2)/2 )
= ( 1/2, -4/2 )
= ( 1/2, -2 )
4. Find the midpoint of DA:
Midpoint DA = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
DA = ( ( -4 + -4)/2, (3 - -2)/2 )
= ( -8/2, 5/2 )
= ( -4, 5/2 )
5. Connect the midpoints to form a new figure.
The lines connecting the midpoints are: AB - BC - CD - DA.
The lines of reflection that flipped the given figure onto itself are the perpendicular bisectors of these connecting lines.
Explain Bot
answered
To determine the lines of reflection that flipped the given figure onto itself, we need to examine the reflection properties of the figure.
Let's start by plotting the given points on a coordinate plane:
A: (-4, 3)
B: (5, 3)
C: (5, -2)
D: (-4, -2)
Plotting these points, we get the following figure:
y
|
|
|
|
|___________________x
-4 -2 0 2 4 6
Now, to find the lines of reflection that flipped the figure onto itself, we need to look for lines that pass through the midpoints of the line segments connecting corresponding points.
Let's find the midpoints of the line segments AB and CD:
Midpoint of AB:
x-coordinate: (-4 + 5)/2 = 0.5
y-coordinate: (3 + 3)/2 = 3
Midpoint of CD:
x-coordinate: (-4 + 5)/2 = 0.5
y-coordinate: (-2 + -2)/2 = -2
Now, let's plot these midpoints on the coordinate plane:
Midpoint of AB: (0.5, 3)
Midpoint of CD: (0.5, -2)
y
|
|
-2
|
-2
|________________________________x
-4 -2 0 2 4 6
As we can see, the two midpoints lie on a horizontal line that passes through the point (0.5, 0). This horizontal line is the line of reflection that flipped the given figure onto itself.
Therefore, the equation for the line of reflection is: y = 0
In addition to the horizontal line, we can also observe that there is a vertical line passing through the figure that serves as another line of reflection. This vertical line runs through the midpoint of the line segment AD. To find the midpoint of AD:
Midpoint of AD:
x-coordinate: (-4 + -4)/2 = -4
y-coordinate: (3 + -2)/2 = 0.5
y
0.5 |
|
|
|
|____________________x
-4 -2 0 2 4 6
As we can see, the midpoint of AD is (-4, 0.5), which lies on a vertical line passing through the figure.
Therefore, the equation for the vertical line of reflection is: x = -4
To summarize, the lines of reflection that flipped the given figure onto itself are:
1. Horizontal line: y = 0
2. Vertical line: x = -4