Similarity
Similarity
- Two triangles are similar if they have the same shape but they may not be the same size.
- Properties of similar triangles are:• all the corresponding angles are equal,• all the three pairs of corresponding sides have the same ratio (proportional).
- The symbol for similarity is \(\sim\).
- Similarity Tests for Triangles
AA Similarity Test : The two triangles are similar if any two angles of one triangle are equal to any two of the corresponding angles of the other triangle.
| \({ \left. \begin{array}{l} \angle{BAC} = \angle{QPR} \\ \angle{ABC} = \angle{PQR} \\ \angle{ACB} = \angle{PRQ} \end{array} \right\} \begin{array}{l} \mathrm{Any} \, \mathrm{two} \, \mathrm{equivalent} \, \\ \mathrm{angles} \, \end{array} }\) \[\therefore \Delta ABC \text{ and}\hspace{0.5em} \Delta PQR \text{ are similar.}\] (\(\text{AA Similarity Test}\)) |
|---|---|
SSS Similarity Test: The two triangles are similar if all the corresponding sides of the two triangles are proportional with a constant \(k\).
| \[\frac{PQ}{AB} = \frac{PR}{AC} = \frac{QR}{BC} = k\] (** You have to calculate the corresponding ratios of the lengths and if ALL give the same constant \(k\), then the two triangles must be similar.) \[\therefore \Delta ABC \text{ and}\hspace{0.5em} \Delta PQR \text{ are similar.}\] (\(\text{SSS Similarity Test}\)) |
SAS Similarity Test: The two triangles are similar if an angle (included angle) of one triangle is equal to an angle (included angle) of the other triangle, and the sides which include the equal angle of both triangles are proportional.
| \(\angle ABC = \angle PQR\) (included angle) \[\frac{PQ}{AB} = \frac{QR}{BC} = k\] (** You have to calculate the two corresponding ratios of the lengths which are "beside" the "included angle" and if they give the same constant \(k\) and the corresponding "included angle" of both the triangles are equal, then the two triangles must be similar.) \[\therefore \Delta ABC \text{ and}\hspace{0.5em} \Delta PQR \text{ are similar.}\] (\(\text{SAS Similarity Test}\)) |
Video Solution:
Video solution locked
A free account saves your progress; this video needs a course plan.
Solution:
Solution locked
680+ questions here come with free worked solutions. A free account saves your questions and progress; this one needs a course plan.
Create a free account to practise similar questions.
Need help? Join our E-Maths tuition for O-Level success.
Learn more