Similarity
A photo and its enlargement have the same shape at different sizes. What exactly has to match, and what does matching let you work out?
Where you are 10 inside
What this area is about
Two figures are similar when one is a scaled copy of the other: matching angles equal and every matching length multiplied by one scale factor k > 0. Fixed ratios let a length you cannot reach, like a tree’s height, come from lengths you can.
How these fit together
Definitions. Similar figures gives the working test for polygons. Dilations make a scaled copy from a center, and similarity transformations define similarity as a dilation plus rigid motions.
Tests for triangles. AA similarity needs two pairs of equal angles; SSS and SAS similarity work from side ratios, SAS with the included angle.
Results. What a dilation does to a line: a line through the center stays put, and when k \ne 1 any other line goes to a parallel one. The triangle proportionality theorem says a line parallel to one side splits the other two sides in one ratio, and the triangle angle bisector theorem says an angle’s bisector splits the opposite side in the ratio of the two sides at that angle. Scale factor and perimeter, area and volume: perimeters scale by k, areas by k^2, and volumes of similar solids by k^3.
Procedure. Proportions in similar triangles finds a missing side.
A path through it
Similar figures first. Dilations come next, before similarity transformations and dilated lines. AA similarity also follows similar figures and precedes SSS and SAS, triangle proportionality and proportions. The angle bisector theorem follows triangle proportionality. Perimeter, area and volume have no fixed place.
Next: proving two triangles similar and sine, cosine and tangent.