Similar triangles are the triangles which have the same shape but their sizes may vary. All equilateral triangles, squares of any side length are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion. We denote the similarity of triangles here by ‘~’ symbol.
If two or more figures have the same shape but their sizes are different then such objects are called Similar figures. Consider a hula hoop and wheel of a cycle, the shapes of both these objects are similar to each other as their shapes are the same.
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Similar Triangles Definition
In the figure given above, two circles C1 and C2 with radius R and r respectively are similar as they have the same shape, but necessarily not the same size. Thus, we can say that C1~ C2.
It is to be noted that, two circles always have the same shape, irrespective of their diameter. Thus, two circles are always Similar.
Triangle is the smallest three-sided polygon. The condition for the similarity of triangles is;
i) Corresponding angles of both the triangles are equal, and
ii) Corresponding sides of both the triangles are in proportion to each other.
In the given figure, two triangles ΔABC and ΔXYZ are similar only if,
i) ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
ii) AB/XY = BC/YZ = AC/XZ
Hence if the above-mentioned conditions are satisfied we can say that ΔABC ~ ΔXYZ
It is interesting to know that if the corresponding angles of two triangles are equal then such triangles are known as equiangular triangles. For two equiangular triangles we state the Basic Proportionality Theorem (better known as Thales Theorem) as follows:
- For two equiangular triangles, the ratio of any two corresponding sides is always the same.
Properties of Similar Triangles
- Both have the same shape but sizes are different
- Each pair of corresponding angles are equal
- The ratio of corresponding sides is the same
Similar Triangle Formula
According to the definition, two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional. Hence, we can find the dimensions of one triangle with the help of another triangle. If ABC and XYZ are two similar triangles then by the help of below-given formulas or expression we can find the relevant angles and side length.
- ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
- AB/XY = BC/YZ = AC/XZ
Once we have known all the dimensions and angles of triangles, it is easy to find the area of similar triangles.
Similar triangles Theorems and Proof
AA (or AAA) or Angle-Angle Similarity
If any two angles of a triangle are equal to any two angles of another triangle then the two triangles are similar to each other.
From the figure given above, if ∠ A = ∠X and ∠C = ∠Z then ΔABC ~ΔXYZ.
From the result obtained, we can easily say that,
AB/XY = BC/YZ = AC/XZ
SAS or Side-Angle-Side Similarity
If the two sides of a triangle are in the same proportion of the two angles of another triangle, and the angle inscribed by the two sides in both the triangle are equal, then two triangles are said to be similar.
Thus, if ∠A = ∠X and AB/XY = AC/DF then ΔABC ~ΔXYZ.
From the congruency, we can state that,
AB/XY = BC/YZ = AC/XZ
SSS or Side-Side-Side Similarity
If all the three sides of a triangle are in proportion to the three sides of another triangle then the two triangles are similar.
Thus, if AB/XY = BC/YZ = AC/XZ then ΔABC ~ΔXYZ.
From this result, we can infer that-
∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
Similar Triangles Example Problem
Q.1: In theΔABC length of the sides are given as AP = 5 cm , PB = 10cm and BC = 20 cm.Also PQ||BC. Find PQ.
Solution: In ΔABC and ΔAPQ, ∠PAQ is common and ∠APQ = ∠ABC (corresponding angles)
⇒ ΔABC ~ ΔAPQ (AA criterion for similar triangles)
⇒ AP/AB = PQ/BC
⇒ 5/15 = PQ/20