Similar triangles are triangles that have the same shape but not necessarily the same size and the area of similar triangles are related by the square of the ratio of their corresponding sides.

In this article, we will learn about what similar triangles are along with a few solved examples and practice questions for better understanding.

What are Similar Triangles?

Similar triangles are triangles that have the same shape but different sizes. In other words, their corresponding angles are equal, and their corresponding sides are in proportion (i.e., their ratios are equal). Formally, two triangles ABC and DEF are similar if and only if:

• Their corresponding angles are congruent (equal).
• The ratios of the lengths of their corresponding sides are equal.

Area of Similar Triangles

When two triangles are similar, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

If two triangles △ABC and △DEF are similar, with a scale factor k (which means corresponding sides are in the ratio k), then the ratio of their areas can be given as:

A₁ / A₂ = (k/l)²

where

• A₁ and A₂ are the area of △ABC and △DEF
• k and l are the scale factors of corresponding sides

Areas of Similar Triangles Practice Questions

Question 1: △ABC is similar to △DEF with a scale factor of 3/2. If the area of △ABC is 36 square units, what is the area of △DEF?

Solution:

The given triangles, △ABC and △DEF are similar and their scale factor = 3/2

Hence by relation we have:

A₁ / A₂ = (k/l)²

Here A₁ = 36 square units and k/l = 3/2. Putting the given values we get:

36/A₂ = (3/2)²

A₂ = 36 × (4/9)

A₂ = 16 square units.

So, the area of △DEF is 16 square units.

Question 2: If △PQR is similar to △STU, and the sides of △PQR are in the ratio 3:4:5, and its area is 144 square units. Find the area of △STU, if its sides are in the ratio 6:8:10.

Solution:

Given

△PQR ∼ △STU, sides of △PQR are 3:4:5 and area of △PQR = 144 square units.

Also, the ratio of sides of △STU is 6:8:10

Hence the scale factor of the given triangle is 3/6 or 4/8 or 5/10 = 1/2.

Now the area of △STU can be calculated by:

Area of △PQR/ Area of △STU = (1/2)²

Area of △STU = 144 × 4 = 576

Hence, Area of △STU is 576 square units.

Question 3: △XYZ is similar to △ABC with a scale factor of 4/5. If the area of △ABC is 72 square units, what is the area of △XYZ?

Solution:

The given triangles, △XYZ and △ABC are similar and their scale factor = 4/5

Hence by relation we have:

A₁ / A₂ = (k/l)²

Here area of △ABC i.e. A₂ = 72 square units and k/l = 4/5. Putting the given values we get:

A₁/72 = (4/5)²

A₁ = 16/25 × 72

A₁ = 46.08 square units.

So, the area of △XYZ is 46.08 square units.

Question 4: △ABC is similar to △DEF. The sides of △ABC are in the ratio 1:2:3. If the area of △ABC is 125 square units, what is the area of △DEF, given the ratio of sides of△DEF is 3:6:9 ?

Solution:

Given

△ABC ∼ △DEF, sides of △ABC are 1:2:3 and area of △ABC = 125 square units.

Also, the ratio of sides of △DEF is 3:6:9

Hence the scale factor of the given triangle is 1/3 or 2/6 or 3/9 = 1/3.

Now the area of △DEF can be calculated by:

Area of △ABC/ Area of △DEF = (1/3)²

Area of △DEF = 125 × 9 = 1125

Hence, Area of △DEF is 1125 square units.

Question 5: △LMN is similar to △UVW with a scale factor of 3/5. If the area of △LMN is 72 square units, what is the area of △UVW?

Solution:

The given triangles, △LMN and △UVW are similar and their scale factor = 3/5

Hence by relation we have:

A₁ / A₂ = (k/l)²

Here A₁ = 72 square units and k/l = 3/5. Putting the given values we get:

72/A₂ = (3/5)²

A₂ = 72 × (25/9)

A₂ = 200 square units.

So, the area of △UVW is 200 square units.

Practice Problems on Area of Similar Triangles

Problem 1: Triangle ABC is similar to triangle DEF with a scale factor of 2:3. If the area of triangle ABC is 72 square units, what is the area of triangle DEF?

Problem 2: Triangle PQR has sides in the ratio 3:4:5. It is similar to triangle XYZ, whose sides are in the ratio 6:8:10. If the area of triangle PQR is 24 square units, what is the area of triangle XYZ?

Problem 3: △ABC is similar to △DEF. The lengths of corresponding sides are in the ratio 5:2. If the area of △ ABC is 50 square units, what is the area of △DEF?

Problem 4: Triangle MNO is similar to triangle XYZ with a scale factor of 4:5. If the area of triangle MNO is 100 square units, what is the area of triangle XYZ?

Problem 5: Triangle LMN is similar to triangle UVW. If the lengths of corresponding sides are in the ratio 1:3, and the area of triangle LMN is 36 square units, what is the area of triangle UVW?

Problem 6: △PQR is similar to △XYZ. The sides of △PQR are in the ratio 2:3:4, and its area is 36 square units. Find the area of △XYZ if its sides are in the ratio 4:6:8.

Problem 7: △ ABC and △XYZ are similar triangles with a scale factor of 3/4. If the area of △XYZ is 48 square units, Find the area of △ABC.

Problem 8: Triangle LMN is similar to triangle UVW. The scale factor of these triangles is 5:7. Find the area of triangle UVW, if the area of triangle LMN is 100 square units.

Also Read,

• Area of Rectangle Questions
• Practice Problem on Surface Area and Volume Formulas
• Practice Questions on Congruence of Triangles

FAQs on Area of Similar Triangles

What is the relationship between the areas of similar triangles?

The area of similar triangles is proportional to the square of the ratio of their corresponding sides.

What happens to the area of similar triangle if all its sides are doubled?

When all sides of a triangle are doubled, the area of the triangle increases by a factor of 4. This is because the area of a triangle is proportional to the square of its side lengths.

How do similar triangles used in real-world applications?

Similar triangles are used in various real-world applications such as map scaling, architecture, and engineering. They allow us to determine dimensions and distances indirectly, using the principles of similarity and proportionality.

Is there a shortcut to find the area of similar triangles?

Yes, the shortcut involves using the square of the ratio of the corresponding sides. Once you know the ratio of the sides and the area of one triangle, you can quickly find the area of the other triangle using the area ratio formula:

A₁ / A₂ = (k/l)²