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Last updated on October 6, 2025
In geometry, similar triangles are triangles that have the same shape but may differ in size. This means they have equal corresponding angles and proportional corresponding side lengths. In this topic, we will learn the formulas related to similar triangles.
The properties of similar triangles are crucial in geometry. Let’s learn the formulas to identify and calculate the corresponding sides and angles of similar triangles.
Triangles are similar if they satisfy certain criteria:
1. AA Criterion (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
2. SSS Criterion (Side-Side-Side): If the corresponding sides of two triangles are in proportion, the triangles are similar.
3. SAS Criterion (Side-Angle-Side): If one angle of a triangle is equal to one angle of another triangle and the lengths of the sides including these angles are proportional, the triangles are similar.
In similar triangles, the corresponding sides are proportional. The proportionality can be expressed using the formula: \([ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} ] w\)here \((a_1, b_1, c_1) \)are the sides of one triangle and\( (a_2, b_2, c_2) \)are the sides of the other triangle.
The area of similar triangles is proportional to the square of the ratio of their corresponding side lengths. The formula for the ratio of the areas is: \([ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{a_1}{a_2}\right)^2 ]\)
Similar triangles are fundamental in geometry and are used to solve complex problems. Here are some significant uses:
Identifying similar triangles can be tricky. Here are some tips: -
Students often make errors when working with similar triangles. Here are some mistakes and the ways to avoid them.
If triangle ABC is similar to triangle DEF and AB = 4 cm, BC = 6 cm, and DE = 8 cm, find EF.
EF = 12 cm
Since the triangles are similar, the corresponding sides are proportional.
\([ \frac{AB}{DE} = \frac{BC}{EF} \implies \frac{4}{8} = \frac{6}{EF} \implies EF = \frac{6 \times 8}{4} = 12 \text{ cm} ]\)
Two similar triangles have sides in the ratio 3:4. If the area of the smaller triangle is 27 cm², find the area of the larger triangle.
The area of the larger triangle is 48 cm².
The ratio of the areas of similar triangles is the square of the ratio of their sides.
\( [ \frac{\text{Area}_1}{\text{Area}_2} = \left(\frac{3}{4}\right)^2 \implies \frac{27}{\text{Area}_2} = \frac{9}{16} \implies \text{Area}_2 = \frac{27 \times 16}{9} = 48 \text{ cm}^2 ]\)
In similar triangles, the side lengths are in the ratio 5:7. If the shortest side of the larger triangle is 21 cm, find the length of the corresponding side in the smaller triangle.
The corresponding side in the smaller triangle is 15 cm.
Using the side ratio: \([ \frac{5}{7} = \frac{x}{21} \implies x = \frac{5 \times 21}{7} = 15 \text{ cm} ]\)
Triangle XYZ is similar to triangle UVW. If XY = 9 cm, YZ = 12 cm, and UV = 6 cm, find VW.
VW = 8 cm
Using the proportionality of corresponding sides:\([ \frac{XY}{UV} = \frac{YZ}{VW} \implies \frac{9}{6} = \frac{12}{VW} \implies VW = \frac{12 \times 6}{9} = 8 \text{ cm} ]\)
The sides of two similar triangles are in the ratio of 2:3. If the perimeter of the smaller triangle is 30 cm, find the perimeter of the larger triangle.
The perimeter of the larger triangle is 45 cm.
The ratio of the perimeters of similar triangles is equal to the ratio of their corresponding sides.
\([ \frac{\text{Perimeter}_1}{\text{Perimeter}_2} = \frac{2}{3} \implies \frac{30}{\text{Perimeter}_2} = \frac{2}{3} \implies \text{Perimeter}_2 = \frac{30 \times 3}{2} = 45 \text{ cm} ]\)
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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