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dilation scales

Homotecia
geometrical transformation in which each point A of figure corresponds another A 'so that they are aligned with another (center O) and the ratio k between them is constant: OA / OA' = k

two homothetic figures have their corresponding sides parallel and are proportionate: a / a '= b / b' = k

The homothety preserves angles.
The product of 2 distinct homothety of center and power is another homothety whose center is aligned with the other two. Forman
group:
5 - internal operation: the product of 2 dilation is another dilation.
6 - Is associative.
7 - has neutral element is the power dilation k = 1.
8 - It has symmetrical item.



On the left, a square is transformed into another with the center of dilation at the top or both in a ratio of 7 / 4. Right
a circle is transformed into another from the center O at a rate of 5 / 3. In this case the center of dilation is at the end of an original diameter of the circle, this is an invariant point in the transformation, which the two circles are tangent at this point.


dilation Across several properties are met: that the center projection aligns the points homothetic to him. Homothetic figures that have their sides parallel and retain their angles. A relationship of proportionality between the two homothetic figures. Homothetic figures that are always the same shape, but, generally, of different sizes, so the dilation becomes a direct method to change the scale graphically. Dilation is a homology plane in which the homothetic sides are cut in line at infinity.

The dilation is directly or positive if the two figures were the same side of the center of projection as it is reverse or negative if the opposite occurs, as in this case in which a triangle is transformed into the other from a center by a dilation.
As dilation are detected across all the properties of dilation: the sides are parallel homothetic figures, which change the size of all sides proportionally, which preserves angles, which has each pair of points aligned homothetic the projection center, etc.


Another exercise that can be solved by applying a homothety we have in the following example. Given two straight
ab, determine the direction to be followed by another straight line through the point P is cutting the two lines given to b.
It m makes a triangle, blue in the picture, so that one of its vertices coincide with the given point P, and two vertices are on the lines given to b. It builds another triangle that has sides parallel to the former and contain the vertices also incidents in the lines given to b. This triangle drawn in ocher, the former is homothetic, which means that the vertices of both triangles are aligned with a projection center and the other two vertices and has aligned the two missing corners were cut in center of projection is no other place as the intersection of lines to b. Therefore
the line intersects the other two and passing through the point P is given by the apex of the triangle T ocher, the line cuts the green PT aby go straight P.