Similarity (geometry)

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 triangles 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 and the corresponding angles are congruent.

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

Ratio between areas of Similar Triangles

Similar Triangles The ratio of the areas of 2 similar triangles is equal to the ratio of the square of their corresponding sides. PROOF

PQR ~ ABC

To Prove:

Ar(△ABC)/Ar(△PQR)= (AB/PQ)^2

Construction:

AM⊥BC and PN⊥QR

Ar(△ABC)/Ar(△PQR)= (1/2*BC*AM)/(1/2*QR*PN)

=> Ar(△ABC)/Ar(△PQR)= (BC*AM)/(QR*PN) ---- 1

In △ABM and PQN:

∠B = ∠Q (△ABC ~ △PQR)

∠M = ∠N (Right angles)

△ABM ~ △PQN (By AA similarity)

So,

AM/PN = AB/PQ ---- 2

AB/PQ = BC/QR = AC/PR ---- 3

Replace 2 and 3 in 1:

The AM/PN becomes: AB/PQ

So, now, 1 is:

Ar(△ABC)/Ar(△PQR)= (1/2*BC*AB)/(1/2*QR*PQ)