Similarity Triangle is the topic of International GCSE, O level Mathematics (Specification B) Geometry. We will be covering all math topics gradually.
Similarity Triangle
Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths 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.
Definition
Two triangles are similar if they have the same ratio of corresponding sides and equal pair of corresponding angles.
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.
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 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.
Similar Triangle Example
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 (Similar triangles proportions)
Hence, if the above-mentioned conditions are satisfied, then 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 can 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
- Both have the same shape but sizes may be different
- Each pair of corresponding angles are equal
- The ratio of corresponding sides is the same
Formulas
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, we can find the relevant angles and side lengths.
- ∠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.
See more: