Application of similar triangles in the field of geometry
LikeEver wondered why a similar triangle concept is important when studying geometry in your maths lessons? Similar figures have their sides in proportion and all corresponding angles are the same. This quality proves useful while solving 2D and 3D Geometry problems as well as questions from vector theory. Later this knowledge also becomes handy while studying structural engineering nuances.
It's quite un-intuitive but using this approach one could avoid the use of trigonometry ratios altogether. Having used trigonometry for anything and everyone, one actually loses the ability to spot similar triangles and hence this way of problem solving becomes extinct. Hence, in my practice, from time to time I restrict myself to only use similar figure theory to solve seemingly complex geometry problems.
In my world, I would like to give an analogy of the exercise of Surya Namaskar from the ancient science of Yogasana. Restricting yourself to only similar triangle theory gives the much needed exercise to the brain that it massages different parts of the brain compared to when you use trigonometry ratios. It is like saying it gives your brain a different view of the world of geometry. As if you are watching the same movie but now 10 years later with a different level of maturity and with a completely different perspective towards your life.
In order to prove congruence between two triangles there are various criterias like SSS (Side, Side, Side), SAS (Side, Angle, Side), ASA (Angle, Side, Angle) and RHS (Right Angle, Hypotenuse, Side). One common mistake a newbie makes at this delicate juncture in his/her learning journey at this time is to falsely use AAA (Angle, Angle, Angle) in order to prove the two given triangles are congruent stating that Equi-Angle (= Equilateral) triangles are congruent as well. However, it becomes clear just a few moments later that two Equi-Angle/Equilateral triangles can have all sides measuring 5 cm and 10 cm or any two different measures and hence they are not congruent.
The organic learning then takes place when one realizes that each 5 cm sides are in proportion to the corresponding 10 cm sides, eventually forming a common ratio of 1:2 between all sets of sides of two triangles. The next step of unfolding the above said theory then progresses with drawing two Isosceles or scalene triangles where corresponding angles are the same and then the learner practically measures and uncovers that corresponding sides maintain the same proportion. The concept of similarity between figures can then be extended to any other polygons, eventually learning the concepts of enlargement and shrinking of drawings using magnifying or reducing the scale factor.
For more such meaningful and engaging mathematical conversations, feel free to schedule first maths lesson with me for free. As a private maths tutor, I believe that it is our responsibility to keep trying various ways of explaining a concept to the learner until they get it. If it doesn’t make sense, the mathematics concept is not taught in the right way.