Similar Figures
Similar figures mean when two figures are of the same shape but are of different sizes. In other words, two figures are called similar when they both have a lot of the same properties but still may not be identical. For example, the sun and moon might appear the same size but they are actually different in size. However, we are similar figures since both the figures are circular in nature. This phenomenon is considered as the property of similarity keeping the shape and distance in mind. Let us learn more about this interesting concept by defining similar figures, their role in geometry, and solve a few examples.
| 1. | Similar Figures Definition |
| 2. | Application of Similar Figures |
| 3. | Similarity of Triangles |
| 4. | Similarity of Polygons |
| 5. | Difference Between Similarity and Congruence |
| 6. | FAQs on Similar Figures |
Similar Figures Definition
When two or more objects or figures appear the same or equal due to their shape, this property is known as a similarity or similar figures. When we magnify or demagnify these figures, they always superimpose each other. In geometry, when two shapes such as triangles, polygons, quadrilaterals, etc have the same dimension or common ratio but size or length is different, they are considered similar figures. For example, two circles (of any radii) are of the same shape but different sizes because they are similar. Look at the image below.
The symbol to express similar figures is the same symbol for congruence i.e. "∼" but similar does not mean the same in size. Shapes are also considered to be similar when the ratios of the corresponding sides are equivalent i.e. while dividing each set of corresponding side lengths, the number derived is the scale factor. This number helps in increasing or decreasing the figures in size but not in shape leaving them looking like similar figures. For example, a rectangle has a length of 5 units and a width of 2 units. Now, if we increase the size of this rectangle by a scale factor of 2, the sides will become 10 units and 4 units, respectively. Hence, we can use the scale factor to get the dimensions of the changed figures.
Application of Similar Figures
Some applications of similarity or similar figures are mentioned below.
- The similarity is widely used in Architecture.
- Solving problems involves height and distance.
- Solving Mathematical problems involving triangles.
Similarity of Triangles
Two triangles will be similar if the angles are equal (corresponding angles) and sides are in the same ratio or proportion(corresponding sides). Similar triangles may have different individual lengths of the sides of triangles but their angles must be equal and their corresponding ratio of the length of the sides or scale factor must be the same. If two triangles are similar that means,
- All corresponding angle pairs of triangles are equal.
- All corresponding sides of triangles are proportional.
Let us understand the similarity of triangles with the three theorems according to their angles and sides.
AA Similarity Criterion
The AA criterion for triangle similarity states that if the three angles of one triangle are respectively equal to the three angles of the other, then the two triangles will be similar. In short, equiangular triangles are similar. Ideally, the name of this criterion should then be the AAA(Angle-Angle-Angle) criterion, but we call it as AA criterion because we need only two pairs of angles to be equal - the third pair will then automatically be equal by the angle sum property of triangles.
Consider the following figure, in which ΔABC and ΔDEF are equi-angular,i.e.,
- ∠A = ∠D
- ∠B = ∠E
- ∠C = ∠F
Using the AA criterion, we can say that these triangles are similar.
SSS Similarity Criterion
The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar. This essentially means that any such pair of triangles will be equiangular(All corresponding angle pairs are equal) also. Consider the following figure, in which the sides of two triangles ΔABC and ΔDEF are respectively proportional:
That is, it is given that:
\[\frac{{AB}}{{DE}} = \frac{{BC}}{{EF}} = \frac{{AC}}{{DF}}\]
SAS Similarity Criterion
The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar. Note the emphasis on the word included. If the equal angle is a non-included angle, then the two triangles may not be similar. Consider the following figure:
It is given that
\[\begin{align}& \frac{{AB}}{{DE}} = \frac{{AC}}{{DF}} \end{align}\]
∠A = ∠D
The SAS criterion tells us that ΔABC ~ ΔDEF
Similarity of Polygons
Similar polygons have the same shape, whereas their sizes are different. There would be certain uniform ratios in similar polygons. In other words, the corresponding angles are congruent, but the corresponding sides are proportional. Polygons are two-dimensional shapes composed of straight lines. They are said to have a closed shape as all the lines are connected. There are two crucial properties of similar polygons:
- The corresponding angles are equal/congruent. (Both interior and exterior angles are the same)
- The ratio of the corresponding sides is the same for all sides. Hence, the perimeters are different.
Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same (the fourth angles automatically become the same as the interior angle sum is 360 degrees), and two adjacent sides have equal ratios.
Difference Between Similarity and Congruence
The words similarity and congruence are associated with shape and size in geometry. Congruence means the same structure, size, and shape whereas similar figures mean the same shape but different size. Let us look at the difference between both terms.
| Congruence | Similarity |
|---|---|
| Congruence or congruent is referred to figures with the same shape and size. | Similarity or similar figures is referred to figures with the same shape but different sizes. |
| Congruent figures are equal in dimension and superimpose each other. | Similar figures are identical but cannot superimpose each other. |
| Congruence follows theorems of similarity. | Similarity does not follow any theorems. |
| Congruence can be expressed as superimposed and coincidental. | Similarity can be expressed as figures that are similar in nature. |
Important Notes
- If two angles of two triangles are equal then their third angle is always equal.
- The angle bisector of a triangle always divides the triangle into two similar triangles. (Angle Bisector Theorem)
- If two similar triangles have sides in ratio \(\frac{x}{y}\) then the ratio of their areas will be \(\frac{x^2}{y^2}\)
Related Topics
Listed below are a few topics related to similar figures, take a look.
Similar Figures Examples
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Example 1: Consider the following figure:
Find the value of ∠E.
Solution:
Match the longest side with the longest side and the shortest side with the shortest side and check all three ratios. We note that the three sides of the two triangles are respectively proportional:
\[\begin{align}& \left\{ \begin{gathered}\frac{{DE}}{{AB}} = \frac{{4.2}}{6} = 0.7\\ \frac{{DF}}{{AC}} = \frac{{2.8}}{4} = 0.7\\ \frac{{EF}}{{BC}} = \frac{{3.5}}{5} = 0.7 \end{gathered} \right.\\&\quad\frac{{DE}}{{AB}} = \frac{{DF}}{{AC}} = \frac{{EF}}{{BC}} \end{align}\]
Thus, by SAS similarity criterion, ΔABC ~ ΔDEF
This means that they are also equiangular. Note carefully that the equal angles will be:
∠A = ∠D = 55.77°
∠C = ∠F = 82.82°
∠B = ∠E
Finally,
∠E = ∠B = 180° - (55.77° + 82.82°)
∠E = 41.41°
Therefore, ∠E = 41.41°.
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Example 2: Consider two similar triangles, ΔABC and ΔDEF:
AP and DQ are medians in the two triangles respectively. Show that
\[\frac{{AP}}{{BC}} = \frac{{DQ}}{{EF}}\]
Solution:
Since the two triangles are similar, they are equiangular. This means that, ∠B = ∠E
Also,
\[\begin{align} \frac{{AB}}{{DE}} &= \frac{{BC}}{{EF}}\\ \Rightarrow \quad\frac{{AB}}{{DE}} &= \frac{{BC/2}}{{EF/2}} = \frac{{BP}}{{EQ}} \end{align}\]
Hence, by the SAS similarity criterion, ΔABP ~ ΔDEQ
Thus, the sides of these two triangles will be respectively proportional, and so:
\[\begin{align} \frac{{AB}}{{DE}} &= \frac{{AP}}{{DQ}}\\ \Rightarrow \quad\frac{{AP}}{{DQ}} &= \frac{{BC}}{{EF}}\\ \Rightarrow \quad\frac{{AP}}{{BC}} &= \frac{{DQ}}{{EF}} \end{align}\]
Therefore, \[\frac{{AP}}{{BC}} = \frac{{DQ}}{{EF}}\].
FAQs on Similar Figures
How is Similarity Used in Real Life?
The similarity is used in designing, solving problems involving height and distance, etc.
What are the Rules of Similarity?
The three rules of similarity are SSS similarity, SAS similarity, and AA or AAA similarity.
Is SSA a Similarity Theorem?
No, SSA is not a similarity theorem.
What is a Similarity Statement?
When two or more objects or figures appear the same or equal due to their shape, this property is known as a similarity.
What is a SSS Similarity Theorem?
The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar.
What is a SAS Similarity Theorem?
The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.
What is a AA Similarity Theorem?
The AA criterion for triangle similarity states that if the three angles of one triangle are respectively equal to the three angles of the other, then the two triangles will be similar.
What are Similar Polygons?
Two polygons are similar when the corresponding angles are equal/congruent, and the corresponding sides are in the same proportion.