From Earth, the sun and the moon seem to be exactly the same size. Although the sun dwarfs the moon, this appears to happen because both spheres are similar figures!
Explore our app and discover over 50 million learning materials for free.
Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken
Jetzt kostenlos anmeldenNie wieder prokastinieren mit unseren Lernerinnerungen.
Jetzt kostenlos anmeldenFrom Earth, the sun and the moon seem to be exactly the same size. Although the sun dwarfs the moon, this appears to happen because both spheres are similar figures!
This article will cover the definition of similarity in geometry and its applications.
Similarity can be defined as an attribute exhibited by two or more figures when their shapes are the same.
An individual is up for a red-night game with his friends requiring them to blindfold each other and make a selection for a similar pair among 4 pastries; a doughnut, a burger bread, sliced bread, and a samosa. Determine the appropriate selection.
Solution:
A samosa is triangular in shape; a slice of bread is rectangular in shape; a burger bread is circular in shape, and a doughnut is circular in shape.
Hence the appropriate similar pair is the burger bread and doughnut.
Two figures are said to be similar if they have the same shape but different sizes. Therefore, similar shapes have the following properties,
To have a deep understanding of the application of similarity in geometry, present the similarity theorems for triangles.
Many people (perhaps, you) get excited when there is a formula to solve problems in a topic. These formulas become the identity of these topics and serve to enhance memory retention. However, the concept of similarity lacks this approach. In clearer terms, there is hardly any formula(s) that is attributed to solving the similarity problems.
Nonetheless, problems with similarity are primarily reliant on the understanding and application of the properties of similarity which was discussed in the previous section. More so and more importantly, the understanding, as well as the application of the theorems which are discussed hereafter are indeed much more than having formulas to memorize.
There are multiple ways in which we can determine whether or not two triangles are similar, by using one of the four triangle theorems.
If two angles in a triangle are equal to two angles in another triangle, then these two triangles are similar.
Triangles ABC and DEF are similar, since
and
If an angle in one triangle is equal to an angle in another triangle and the sides making up this angle are proportional, then these two triangles are similar.
What is meant by proportionality of sides, is that the two sides on triangle ABC must both be multiplied by the same number to give the sides of triangle DEF.
The given sides in the above figure have a common ratio, that is,
and the respective angles formed by these corresponding sides are equal,
Two triangles could also be classified as being similar in the event that their sides AC, AB, and BC which correspond to the sides of another triangle DF, DE and FE are indeed proportional.
In the diagram, all of the lines forming triangle DEF are the length of their respective side in triangle ABC multiplied by a constant factor r.
This theorem is valid only for right-angled triangles.
Two triangles are similar if the length of the hypotenuse and one other side in one triangle are proportional to the length of the hypotenuse and the other side in another triangle. That is
When we use a side in a similarity theorem (for example in the SAS theorem), we do not mean that the sides are equal, but that the ratio between the triangle's sides is constant.
The symbol we use to show that two things are similar is ∼ . Suppose triangles ABC and DEF are similar, we could then write
Δ ABC ∼ Δ DEF.
Triangle ABC has sides AB = 6 cm, AC = 4 cm, and BC = 10 cm. Triangle DEF has sides DE=3 cm, DF = 2 cm, and EF = 5 cm. Prove that these triangles are similar.
Solution:
Since we are only given sides, we want to use the SSS similarity theorem.
For us to be able to apply this theorem, we need to find a common ratio between the sides of triangle ABC and triangle DEF.
The ratio between the sides AB and DE is
The ratio between sides AC and sides DF is
The ratio between sides BC and EF isSince the ratio between the sides of triangle ABC to its respective sides on triangle DEF is constant, we can say that
Polygons are plane shapes that have three or more sides. This means that a triangle is also a polygon. The concept of similarity also occurs in other polygons other than triangles.
In fact, the similarity of triangles is a particular case of the similarity of polygons.
However, for similarity to occur between polygons, two conditions need to be met:
1. The corresponding angles of the pair in comparison must be equivalent.
2. The corresponding sides of the pair in comparison must have equivalent proportions.
Prove that these two rectangles are similar.
Solution:
Both rectangles have all their internal angles as right angles. This means that the first criterion which says that all corresponding angles must be equal has been met.
Next, we need to confirm that the ratio of their corresponding sides is equal.
The ratio of both widths is
and the ratio of both lengths is
Determine similarity among the following pairs,
(a)
(b)
(c)
(d)
Solution:
(a) Using the angle-angle rule, we can tell that both triangles in figure (a) are similar because knowing that the sum of angles in a triangle is 180º, hence the third angle in the first triangle is
This confirms that both triangles are similar since all corresponding angles are equal.
(b) The pair in figure (b) are not similar, although the ratio between corresponding sides is equal to 2:1, the corresponding angles between them are different and as such using the side-angle-side rule we can confirm that the triangular pair are not similar.
(c) The pair in figure (c) are not similar because the ratio of the two sides is 2:1 while the ratio of the third side is 5:3. By considering the side-side-side rule, the ratio of all corresponding sides must be equivalent, hence this pair of triangles are not similar.
(d) The pair in figure (d) is similar because they are both right triangles and the ratio of both corresponding hypotenuses and opposite sides is 1:4. This is in compliance with the right angle-hypotenuse-side rule of similarity.
Two figures are similar if they have the same shape.
Two major rules are considered for similarity; the shape between objects in comparison be the same and the proportion of their corresponding sides must be equivalent.
Similar shapes have equal corresponding angles and proportional sides.
There is no exact formula for calculating similarity in geometry, however, the similarity is determined based on compliance with some rules or theorems.
An example of similarity in geometry is two triangles that have equivalent corresponding angles and the proportion of their corresponding sides is the same for all three sides.
Define similarity.
Similarity in geometry is when two shapes can be the same after being flipped, turned, or stretched
What are the four triangle similarity theorems?
RHS, SSS, SAS, AA
When two polygons have equivalent sides, then, what other factor would determine that they are similar?
Their corresponding sides but be equivalent in proportion.
What properties certify that two figures are similar?
Corresponding angles must be equal and corresponding sides are in the same ratio.
All of the following are theorems which determine similarity in triangles except?
Side-Side
If an angle in one triangle is equal to an angle in another triangle and the sides making up this angle are proportional, then these two triangles are similar. What theorem proves this?
Side-Angle-Side
Already have an account? Log in
Open in AppThe first learning app that truly has everything you need to ace your exams in one place
Sign up to highlight and take notes. It’s 100% free.
Save explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.
Already have an account? Log in
Already have an account? Log in
The first learning app that truly has everything you need to ace your exams in one place
Already have an account? Log in
Similarity in geometry examples
To further understand the concept of similarity, here are a few examples.