Similarity Transformations
Related Topics:
Common Core (Geometry)
Common Core for Mathematics
Use transformations to prove similarity.
Similarity Transformations
The Common Core State Standards (CCSS) videos are designed to support states, schools, and teachers in the implementation of the CCSS. Each video is an audiovisual resource that focuses on one or more specific standards and usually includes examples/illustrations geared to enhancing understanding. The intent of each content-focused video is to clarify the meaning of the individual standard rather than to be a guide on how to teach each standard although the examples can be adapted for instructional use.
Geometry2D - Similarity
we can define similarity of two geometrical objects on a plane as possibility to transform one into another using dilation (scaling), as defined above, optionally combined with congruent transformations of parallel shift (translation), rotation and symmetry (reflection). Geometry2D - Similarity - Triangles
Recall that we have defined similarity as the characteristic of one geometrical object to be an image of another after transformation of scaling and, possibly, some congruent transformation (translation, rotation and symmetry relative to an axis). Applied to triangles, we see, first of all, that an image of a triangle after transformation of scaling is a triangle (since straight lines are transformed into straight lines). We also observe that similar triangles have corresponding angles congruent (since scaling (dilation) preserves angles) and corresponding sides proportional with the same coefficient of proportionality equal to a scaling factor (since transformation of scaling changes the lengths of all segments by the same scaling factor). All in all, scaling transforms a triangle into another triangle, similar to original (by definition of similarity), with correspondingly congruent angles and correspondingly proportional sides.
The property of triangles to have congruent angles and proportional sides is, actually, equivalent to their similarity. In fact, three much shorter statements are true, each one, obviously, necessary and, as we are going to prove, sufficient conditions for similarity of triangles.
Theorem 1. If two triangles have two pairs of angles correspondingly congruent to each other, then they are similar.
Theorem 2. If two triangles have one pairs of congruent angles and sides, forming these angles, are proportional, then they are similar.
Theorem 3. If three sides of one triangle are correspondingly proportional to three sides of another triangle, then these triangles are similar. Geometry - Similarity - Circles
Theorem 1 Scaling of a circle results in a circle. The radii of these circles are related by a scaling factor.
Theorem 2 Any two circles are similar.
Theorem 3 For any two circles, positioned outside of each other, there is a transformation of scaling that transforms one into the other. Similarity of Circles
How to prove all circles are similar? Quadrilateral similarity by showing congruent angles
Common Core (Geometry)
Common Core for Mathematics
Examples, solutions, videos, and lessons to help High School students learn when given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Common Core: HSG-SRT.A.2
Similarity and TransformationsUse transformations to prove similarity.
The Common Core State Standards (CCSS) videos are designed to support states, schools, and teachers in the implementation of the CCSS. Each video is an audiovisual resource that focuses on one or more specific standards and usually includes examples/illustrations geared to enhancing understanding. The intent of each content-focused video is to clarify the meaning of the individual standard rather than to be a guide on how to teach each standard although the examples can be adapted for instructional use.
we can define similarity of two geometrical objects on a plane as possibility to transform one into another using dilation (scaling), as defined above, optionally combined with congruent transformations of parallel shift (translation), rotation and symmetry (reflection). Geometry2D - Similarity - Triangles
Recall that we have defined similarity as the characteristic of one geometrical object to be an image of another after transformation of scaling and, possibly, some congruent transformation (translation, rotation and symmetry relative to an axis). Applied to triangles, we see, first of all, that an image of a triangle after transformation of scaling is a triangle (since straight lines are transformed into straight lines). We also observe that similar triangles have corresponding angles congruent (since scaling (dilation) preserves angles) and corresponding sides proportional with the same coefficient of proportionality equal to a scaling factor (since transformation of scaling changes the lengths of all segments by the same scaling factor). All in all, scaling transforms a triangle into another triangle, similar to original (by definition of similarity), with correspondingly congruent angles and correspondingly proportional sides.
The property of triangles to have congruent angles and proportional sides is, actually, equivalent to their similarity. In fact, three much shorter statements are true, each one, obviously, necessary and, as we are going to prove, sufficient conditions for similarity of triangles.
Theorem 1. If two triangles have two pairs of angles correspondingly congruent to each other, then they are similar.
Theorem 2. If two triangles have one pairs of congruent angles and sides, forming these angles, are proportional, then they are similar.
Theorem 3. If three sides of one triangle are correspondingly proportional to three sides of another triangle, then these triangles are similar. Geometry - Similarity - Circles
Theorem 1 Scaling of a circle results in a circle. The radii of these circles are related by a scaling factor.
Theorem 2 Any two circles are similar.
Theorem 3 For any two circles, positioned outside of each other, there is a transformation of scaling that transforms one into the other. Similarity of Circles
How to prove all circles are similar? Quadrilateral similarity by showing congruent angles
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