Translate the point of reflection as well as the object such that the point of reflection is mapped to the origin.Find the coordinates of the point of reflection.We can however use a similar strategy as described above for reflections over a horizontal or vertical line. In other words, the ordered pair (x, y) becomes (-x, -y).Īlthough the base concept remains the same for any point, in practice, if the point of reflection is not the origin, we cannot simply change the sign of the x- and y-coordinates to find the coordinates of the reflection. This essentially means changing the sign of every x- and y-value to its opposite sign. Reflection about the origin can be thought of as reflecting across the x-axis and then the y-axis (or vice versa). The simplest and most commonly used point is the origin. Objects can also be reflected over any point in the coordinate plane. It is possible to mostly perform the reflection by sight with the understanding that the reflected object will lie an equal distance from the line of reflection as the original.
This is just one way to reflect objects over a horizontal or vertical line. Referencing the example in step 3, we would shift the reflected object 2 units to the left after reflecting it.
Undo the translation that mapped the horizontal or vertical line to the corresponding axis. Perform the reflection as we would in the x-axis or y-axis reflection examples above. For example, given the vertical line y = -2, translating the line (as well as the object) 2 units to the right would map the line to the y-axis. Translate the line and object such that a horizontal line is mapped to the x-axis, or a vertical line is mapped to the y-axis. Consider the line of reflection and object being reflected to be a single object. To reflect objects over horizontal or vertical lines that are not the x- or y-axes, the following steps can be used: Reflection over a horizontal or vertical line All of the points on triangle ABC undergo the same change to form DEF. Triangle ABC is reflected across the line y = x to form triangle DEF. Reflection over y = xĪ reflection across the line y = x switches the x and y-coordinates of all the points in a figure such that (x, y) becomes (y, x). Triangle DEF is formed by reflecting ABC across the y-axis and has vertices D (4, -6), E (6, -2) and F (2, -4). Algebraically, the ordered pair (x, y) becomes (-x, y). In a reflection about the y-axis, the y-coordinates stay the same while the x-coordinates take on their opposite sign. Triangle DEF is formed by reflecting ABC across the x-axis and has vertices D (-6, -2), E (-4, -6) and F (-2, -4). Algebraically, the ordered pair (x, y) becomes (x, -y). In a reflection about the x-axis, the x-coordinates stay the same while the y-coordinates take on their opposite signs. The most common cases use the x-axis, y-axis, and the line y = x as the line of reflection. There are a number of different types of reflections in the coordinate plane. This is true for any corresponding points on the two triangles and this same concept applies to all 2D shapes. A, B, and C are the same distance from the line of reflection as their corresponding points, D, E, and F. The figure below shows the reflection of triangle ABC across the line of reflection (vertical line shown in blue) to form triangle DEF. The same is true for a 3D object across a plane of refection. In a reflection of a 2D object, each point on the preimage moves the same distance across the line of reflection to form a mirror image of itself. The term "preimage" is used to describe a geometric figure before it has been transformed "image" is used to describe it after it has been transformed. When an object is reflected across a line (or plane) of reflection, the size and shape of the object does not change, only its configuration the objects are therefore congruent before and after the transformation. In geometry, a reflection is a rigid transformation in which an object is mirrored across a line or plane. 
Home / geometry / transformation / reflection ReflectionĪ reflection is a type of geometric transformation in which a shape is flipped over a line.
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