For now it is probably easiest to say that we should be able to move one shape, and if necessary turn it through an angle or flip it over, so that it lies exactly over the other shape, with all corresponding points having the same x and y coordinates.Ĭongruent triangles with different coordinates and orientation Each of these terms represents some action that can be applied to a shape in order to change its location or orientation, and they will be discussed fully elsewhere. For two shapes to be congruent, it must be possible to map one shape exactly onto the other by using a sequence of translations, rotations and reflections. For any two-dimensional shape, the points that define that shape can be viewed as a mapping, because they can be specified as a set of x and y coordinates. A cube is defined by eight points in a three-dimensional space, while a sphere is defined by a single point in a three-dimensional space and all of the points that lie a given distance from that point in any direction.įor the remainder of this discussion, we will concern ourselves only with two-dimensional shapes, although the same principles can be applied equally well to three-dimensional objects.
In fact, any three points on a plane can define a triangle, providing they do not all fall on the same line.
Likewise, a triangle is a two-dimensional shape that is defined by three points on a plane. A circle, for example, is a two dimensional shape that has a specific centre (or origin), and a perimeter consisting of all of the points that are a given distance (the radius) from the centre on the same plane. Two congruent three-dimensional shapes may co-exist in the same three-dimensional space, but have different spatial coordinates and may be oriented differently around the x, y and z axes.Īny simple two or three-dimensional shape may be specified in terms of the x, y and (for three dimensional shapes) z coordinates of the points that define its boundaries.
Two congruent two-dimensional shapes, for example, may co-exist on the same plane, or be on different planes. In other words, they have the same shape and size, but often do not share the same location or orientation. In geometry, if two objects are described as being congruent, the implication is that one can be mapped exactly to the other. The word congruence is derived from the Latin word congruo, which essentially means I agree.
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