Difference between revisions of "Similarity (geometry)"

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More formally, we say two objects are [[congruent (geometry)|congruent]] if they are the same up to translation, rotation and reflection ([[rigid motion]]s).  We say two objects are similar if they are congruent up to a [[dilation]].
 
More formally, we say two objects are [[congruent (geometry)|congruent]] if they are the same up to translation, rotation and reflection ([[rigid motion]]s).  We say two objects are similar if they are congruent up to a [[dilation]].
  
All circles are similar.  Two [[triangle]]s are similar if they have the same [[angle]]s ([[AA similarity]]), and since the sum of the angles of any triangle is 180 [[degree (geometry) | degrees]], this means that two triangles are similar if they have two equal angles (AA similarity).  Equivalently, two triangles are similar if their corresponding sides are in equal [[ratio]]s.  Two [[polygon]]s are similar if their corresponding angles are equal and corresponding sides are in a fixed ratio.  Note that for polygons with 4 or more sides, both of these conditions are necessary.  For instance, all [[rectangle]]s have the same angles, but not all rectangles are similar.
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==Determining Similarity==
  
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* All circles are similar. 
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* There are three ways of determining if two [[triangle]] are similar.
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** If two of the triangles' corresponding [[angle|angles]] are the same, the triangles are similar by [[AA similarity|AA Similarity]].  Note that by the Triangle Angle Theorem, the third corresponding angle is also the same from the two triangles.
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** Two triangles are similar if all their corresponding sides are in equal [[ratio|ratios]] by SSS Similarity.
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** If two of the triangles' corresponding sides are in equal ratio and the corresponding angle between the two sides are the same the triangles are similar by SAS Similarity.
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* Two [[polygon]]s are similar if their corresponding angles are equal and corresponding sides are in a fixed ratio.  Note that for polygons with 4 or more sides, both of these conditions are necessary.  For instance, all [[rectangle]]s have the same angles, but not all rectangles are similar.
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==Applications to Similarity==
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Once two figures are determined to be similar, the corresponding sides are proportional.
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Similar figures (especially triangles) can be usually found in figures that contain many pairs of equal angles.
  
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[[Category:Geometry]]
 
[[Category:Geometry]]

Revision as of 01:03, 5 June 2019

Informally, two objects are similar if they are similar in every aspect except possibly size or orientation. For example, a globe and the surface of the earth are, in theory, similar.

More formally, we say two objects are congruent if they are the same up to translation, rotation and reflection (rigid motions). We say two objects are similar if they are congruent up to a dilation.

Determining Similarity

  • All circles are similar.
  • There are three ways of determining if two triangle are similar.
    • If two of the triangles' corresponding angles are the same, the triangles are similar by AA Similarity. Note that by the Triangle Angle Theorem, the third corresponding angle is also the same from the two triangles.
    • Two triangles are similar if all their corresponding sides are in equal ratios by SSS Similarity.
    • If two of the triangles' corresponding sides are in equal ratio and the corresponding angle between the two sides are the same the triangles are similar by SAS Similarity.
  • Two polygons are similar if their corresponding angles are equal and corresponding sides are in a fixed ratio. Note that for polygons with 4 or more sides, both of these conditions are necessary. For instance, all rectangles have the same angles, but not all rectangles are similar.

Applications to Similarity

Once two figures are determined to be similar, the corresponding sides are proportional.

Similar figures (especially triangles) can be usually found in figures that contain many pairs of equal angles.

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