Key Properties and Formulas for Similar Triangles
The area of two similar triangles suggests that if two triangles stand similar to each other, then the ratio of areas of similar triangles will be proportional to the square of the ratio of corresponding sides of similar triangles. This proves that the ratio of the area of both the similar triangles is proportional to the squares of the corresponding sides of the two similar triangles. The similarity of triangles is denoted by the symbol ‘~’.
Properties of Area of Similar Triangles
The two similar triangles have the same shape but may differ in sizes.
The ratio of corresponding sides of similar triangles is the same.
Each pair of corresponding angles of similar triangles are equal.
Formulas of Area of Similar Triangles
As per the definition, two triangles are known to be similar if their corresponding sides are proportional and corresponding angles are congruent. Thus, we can determine the dimensions of one triangle using another triangle. If PQR and XYZ are two similar triangles, then using the below-given formulas, we can simply identify the relevant side lengths and angles.
∠P = ∠X, ∠Q = ∠Y and ∠R = ∠Z
PQ/XY = QR/YZ = PR/XZ
Once we get familiar with all the angles and dimensions of triangles, it is easy to identify the area of similar triangles.
Similar Triangles and Congruent Triangles
Below is the comparison of similar triangles and congruent triangles in the tabular form.
Similar Triangles Theorems with Proofs
There are various theorems that we will now learn. These are basically used to solve the problems surrounded on similar triangles along with the proofs for each.
1. Angle-Angle Similarity or (AAA)
By the principle of AAA, it suggests that if any two angles of a triangle are in equivalence to any two angles of another triangle, then the two triangles will be similar to each other.
From the figure given below, if ∠ A = ∠D and ∠C = ∠F then ΔABC ~ΔDEF.
From the outcome we attained, we can easily conclude that,
AB/DE = BC/EF = AC/DF
And ∠B = ∠Y
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2. Side-Side-Side Similarity (SSS)
By the postulation, if all the three sides of a triangle are given in a proportion to the three sides of another triangle, then the two triangles are said to be similar.
Thus, if AB/DE = BC/EF = AC/DF then ΔABC ~ΔDEF.
From the above outcome we obtained, we can come to the conclusion that-
∠A = ∠D, ∠B = ∠E and ∠C = ∠F
3. Side-Angle-Side Similarity or (SAS)
By the postulation, it is implied that if the two sides of a triangle or a similar object are in the same proportion of the two sides of the another triangle, and the angle carved out by the two sides in both the triangles are equivalent to one another, then two triangles are said to be similar.
Therefore, if ∠A = ∠D and AB/DE = AC/DF then ΔABC ~ΔDEF.
From the principle of congruence,
AB/DE = BC/EF = AC/DF
and ∠B = ∠E and ∠C = ∠F
Solved Examples
Let's consider an example to understand the similar triangles and congruence in a better way.
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Example:
In the ΔABC the length of the sides are given as AP = 10 cm , PB = 15 cm and BC = 30 cm. Also the side PQ||BC. Determine the PQ.
Solution:
In the given ΔABC and ΔAPQ,
∠PAQ is common and ∠APQ = ∠ABC (using the corresponding angles)
⇒ ΔABC ~ ΔAPQ (By the principle of AAA criterion for similar triangles)
⇒ AP/AB = PQ/BC
⇒ 10/20 = PQ/30
⇒ PQ = 20/2 cm
⇒ PQ = 10 cm
Example:
Check if the two triangles namely ΔABC and ΔDEF, are similar. The two angle given in triangle ABC are ∠A = 40° and ∠B = 70° while for triangle DEF are ∠D=60° and ∠F=80°.
Solution:
In triangle ABC, using angle sum property;
∠A + ∠B + ∠C = 180°
40° + 70° + ∠C = 180°
110° + ∠C = 180°
Subtract both sides by 110°.
∠ C= 70°
Again in triangle DEF, ( by the criterion of angle sum property)
∠D + ∠E + ∠F = 180°
∠60° + ∠E + ∠80°= 180°
∠ 140° + ∠E = 180 °
Now, Subtracting both sides by 140°
we get
∠ E = 40°
Since,∠A = ∠ E = 40° and ∠C = ∠ F= 70°
Thus, by Angle-Angle (AA) rule,
ΔABC~ΔDEF.
FAQs on Area of Similar Triangles Explained
1. What are the main criteria for determining if two triangles are similar?
Two triangles are considered similar if they satisfy any of the following criteria:
AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle.
SAS (Side-Angle-Side): If two sides of one triangle are in proportion to two sides of another triangle, and the included angles are congruent.
SSS (Side-Side-Side): If all three corresponding sides of two triangles are in the same proportion.
These conditions ensure the triangles have the same shape, even if their sizes differ, which is the prerequisite for applying the area theorem.
2. What is the theorem relating the areas of two similar triangles?
The theorem on the area of similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If ΔABC ~ ΔPQR, then:
Area(ΔABC) / Area(ΔPQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)².
3. How do you prove the area of similar triangles theorem as per the NCERT syllabus for 2025-26?
To prove the theorem, you follow these steps:
Given two similar triangles, say ΔABC ~ ΔPQR.
Construct altitudes: Draw AM ⊥ BC and PN ⊥ QR.
Write the area formula for each triangle: Area(ΔABC) = ½ × BC × AM and Area(ΔPQR) = ½ × QR × PN.
Form the ratio of the areas: [Area(ΔABC) / Area(ΔPQR)] = (BC × AM) / (QR × PN).
Prove that the triangles containing the altitudes (ΔABM and ΔPQN) are similar using the AA criterion. This shows that the ratio of corresponding altitudes is equal to the ratio of corresponding sides (AM/PN = AB/PQ).
Since the main triangles are similar, we know AB/PQ = BC/QR. Therefore, AM/PN = BC/QR.
Substitute this result back into the area ratio equation to get: Area Ratio = (BC/QR) × (BC/QR) = (BC/QR)², proving the theorem.
4. Why is the area ratio of similar triangles equal to the square of the side ratio, not just the side ratio itself?
This is because area is a two-dimensional measure, calculated by multiplying two lengths (like base and height). When a triangle is scaled up or down by a factor 'k', both its base and its height are scaled by the same factor 'k'. Consequently, the new area becomes proportional to (k × base) × (k × height), which is k² × (base × height). This is why the area scales by the square of the scaling factor of the sides, not just the factor itself.
5. If two similar triangles have areas of 81 cm² and 144 cm², what is the ratio of their corresponding medians?
First, we find the ratio of the corresponding sides using the area theorem. The ratio of the areas is 81/144. Therefore, the ratio of the sides is the square root of the area ratio:
Ratio of sides = √(81/144) = 9/12 = 3/4.
A key property of similar triangles is that the ratio of their corresponding medians, altitudes, and angle bisectors is the same as the ratio of their corresponding sides. Hence, the ratio of their corresponding medians is also 3:4.
6. How is the area of similar triangles concept applied in real-world scenarios?
The concept of the area of similar triangles has several practical applications, including:
Architecture and Engineering: When creating scaled models of buildings or structures, architects use this principle to calculate the surface area of materials needed for the actual project based on the model's area.
Cartography (Map Making): Maps are scaled-down similar representations of geographical areas. The theorem helps in calculating the actual land area of a region (like a park or a lake) by measuring its area on the map and using the map's scale factor squared.
Photography and Optics: The area of an image projected onto a sensor or film is related to the square of its distance from the lens, a concept derived from the principles of similar triangles.
7. If the ratio of the sides of two similar triangles is 5:7, what is the ratio of their areas?
To find the ratio of the areas, we square the ratio of the corresponding sides.
Given the side ratio is 5:7 or 5/7.
The ratio of the areas will be (5/7)² = 25/49.
Therefore, the ratio of the areas of the two similar triangles is 25:49.
