Similar, in mathematics, is having the same shape but not necessarily the same size (see congruent). Two triangles are similar if they have equal angles and their corresponding sides, say, a₁, b₁, c₁ and a₂, b₂, c₂, have a common ratio, r : a₁/a₂ = b₁/b₂ = c₁/c₂. In general, a similarity is a transformation under which the distance between any corresponding pair of points changes by the same factor.

 

Sherlock Holmes uses the theory of similar triangles to solve a 250-year-old mystery known as the Musgrave Ritual (in a short story of the same name) – an enigmatic series of clues that refers to the shadow of an elm tree when the sun is just visible at the top of a nearby oak to point toward buried treasure. The great detective recalls to Watson his conversation with Reginald Musgrave:

 

"Have you any old elms?" ...
"There used to be a very old one over yonder, but it was struck by lightening ten years ago, and we cut down the stump."
"You can see where it used to be?"
"Oh, yes." ...
"I suppose it is impossible to find out how high the elm was?"
"I can give it you it at once. It was sixty-four feet.... When my old tutor used to give me an exercise in trigonometry, it always took the shape of measuring heights." ...
I went with Musgrave to his study and whittled myself this peg, to which I tied this long string with a knot at each yard. Then I took two lengths of a fishing-rod, which came to just six feet... The sun was just grazing the top of the oak. I fastened the rod on end, marked out the direction of the shadow.... It was nine feet in length. Of course, the calculation was now a simple one. If a rod of six feet threw a shadow of nine, a tree of sixty-four feet would throw one of ninety-six... I measured out the distance ... and I thrust a peg into the spot.

 

Similarity of triangles

Two triangles are called similar if the angles and ratios of corresponding sides of one are equal to those of the other.

 

Similarity of polygons

Polygons are called similar if they agree in respect of all angles and ratios of pairs of sides. The areas of similar polygons vary as the squares of corresponding sides.

 

Similarity transformations

A similarity transformation transforms any plane figure into a similar figure. Conversely, given two similar figures, a similarity transformation can always be found which will send one figure into the other. Similarity transformations belong to the affine transformations.