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Geometric Transformation

Reflection over X Axis

Understanding Reflection across the X-Axis

Reflection across the X-axis is a type of geometric transformation that moves every point of an object to a new position symmetrical to the X-axis. Imagine the X-axis as a flat mirror.

If a point PP has coordinates (x,y)(x,y), then its reflection, which we'll call PP', will have the same xx-coordinate, but its yy-coordinate will be the negative of the original yy value.

Mathematically, if the initial point is P(x,y)P(x,y), then after reflection across the X-axis, its image is P(x,y)P'(x,-y).

Visualizing Points and Their Reflections

Let's observe some points and their reflections after being mirrored across the X-axis.

Notice how the yy-coordinate changes sign, while the xx-coordinate remains the same.

Point Reflections Across the X-Axis
Visualization of points A,B,C,DA, B, C, D and their reflections A,B,C,DA', B', C', D' after mirroring across the X-axis.

Based on the interactive visualization above, we can observe the relationship between the original points (pre-image) and their reflections (image) as follows:

  • Point A(5,2)A(-5,2) becomes A(5,2)A'(-5,-2)
  • Point B(3,1)B(-3,1) becomes B(3,1)B'(-3,-1)
  • Point C(1,2)C(1,2) becomes C(1,2)C'(1,-2)
  • Point D(4,2)D(4,-2) becomes D(4,2)D'(4,2)

The visible pattern is that the xx value remains constant, and the yy value changes sign (becomes its opposite).

Property of Reflection across the X-Axis

Based on the observations above, we can formulate the property of reflection across the X-axis:

P(x,y)X-axisP(x,y)P(x,y) \xrightarrow{\text{X-axis}} P'(x,-y)

This means the image of point P(x,y)P(x,y) reflected across the X-axis is P(x,y)P'(x,-y). The X-axis in this case acts as the line y=0y=0.

Application Examples

Reflecting a Triangle

Determine the image of triangle ABCABC with vertices A(1,4)A(-1,4), B(2,1)B(2,1), and C(2,1)C(-2,-1) reflected across the X-axis.

To determine the image of triangle ABCABC, we apply the reflection property to each of its vertices:

A(1,4)X-axisA(1,4)A(-1,4) \xrightarrow{\text{X-axis}} A'(-1,-4)
B(2,1)X-axisB(2,1)B(2,1) \xrightarrow{\text{X-axis}} B'(2,-1)
C(2,1)X-axisC(2,1)C(-2,-1) \xrightarrow{\text{X-axis}} C'(-2,1)

Consequently, the image of triangle ABCABC is triangle ABCA'B'C' with vertices A(1,4)A'(-1,-4), B(2,1)B'(2,-1), and C(2,1)C'(-2,1).

Reflection of Triangle ABCABC across the X-Axis
Visualization of triangle ABCABC (orange) and its reflection ABCA'B'C' (purple) after mirroring across the X-axis.

Reflecting a Line

If a line has the equation 2x3y=02x - 3y = 0 and is reflected across the X-axis, determine the equation of its reflected line.

Alternative Solution:

Let an arbitrary point P(a,b)P(a,b) lie on the line 2x3y=02x - 3y = 0. Then, the following holds:

2a3b=02a - 3b = 0

The point P(a,b)P(a,b) reflected across the X-axis produces the image P(a,b)P'(a,-b).

To obtain the equation of the reflected line, we substitute the coordinates of the image into new variables. Let x=ax' = a and y=by' = -b.

From this, we get a=xa = x' and b=yb = -y'.

Substitute a=xa = x' and b=yb = -y' into the original equation 2a3b=02a - 3b = 0:

2(x)3(y)=02(x') - 3(-y') = 0
2x+3y=02x' + 3y' = 0

Since xx' and yy' are arbitrary variables representing the coordinates on the reflected line, we can rewrite them as xx and yy.

Thus, the equation of the reflected line is:

2x+3y=02x + 3y = 0
Reflection of Line 2x3y=02x - 3y = 0 across the X-Axis
The original line 2x3y=02x - 3y = 0 (lime green) and its reflection 2x+3y=02x + 3y = 0 (magenta) after mirroring.

This shows how the equation of a line changes after being reflected across the X-axis.