Similar triangles are an important concept in geometry that is studied in Year 9. Two
triangles are said to be similar if their corresponding angles are congruent and their
corresponding sides are in proportion. This means that if we take any two corresponding
sides of the triangles and divide them by each other, the result will be the same for all pairs of
corresponding sides.
The symbol used to indicate that two triangles are similar is ∼. For example, if triangle ABC
is similar to triangle DEF, we write ABC ∼ DEF.
Similar triangles have several important properties that are useful in geometry and real-life
applications. Some of these properties are:
1. Corresponding angles are congruent: When two triangles are similar, their
corresponding angles are congruent. This means that if angle A in triangle ABC is
congruent to angle D in triangle DEF, then angle B in triangle ABC is congruent to
angle E in triangle DEF, and angle C in triangle ABC is congruent to angle F in
triangle DEF.
2. Corresponding sides are in proportion: When two triangles are similar, their
corresponding sides are in proportion. This means that if triangle ABC is similar to
triangle DEF, then:
AB/DE = BC/EF = AC/DF
The ratio of the lengths of any two corresponding sides is equal to the ratio of the
lengths of any other two corresponding sides.
3. The ratio of the perimeters is the same as the ratio of the corresponding sides: If two
triangles are similar, then the ratio of their perimeters is the same as the ratio of the
lengths of their corresponding sides. This means that if triangle ABC is similar to
triangle DEF, then:
(AB + BC + AC) / (DE + EF + DF) = AB/DE = BC/EF = AC/DF
4. The ratio of the areas is the square of the ratio of the corresponding sides: If two
triangles are similar, then the ratio of their areas is equal to the square of the ratio of
their corresponding sides. This means that if triangle ABC is similar to triangle DEF,
then:
Area(ABC)/Area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (AC/DF)^2
5. Similarity transformations: Similar triangles can be obtained by applying similarity
transformations to a given triangle. The three types of similarity transformations are:
a. Reflection: The triangle is reflected across a line, resulting in a mirror image of the
original triangle.
b. Translation: The triangle is moved to a different position without changing its
shape or size.
triangles are said to be similar if their corresponding angles are congruent and their
corresponding sides are in proportion. This means that if we take any two corresponding
sides of the triangles and divide them by each other, the result will be the same for all pairs of
corresponding sides.
The symbol used to indicate that two triangles are similar is ∼. For example, if triangle ABC
is similar to triangle DEF, we write ABC ∼ DEF.
Similar triangles have several important properties that are useful in geometry and real-life
applications. Some of these properties are:
1. Corresponding angles are congruent: When two triangles are similar, their
corresponding angles are congruent. This means that if angle A in triangle ABC is
congruent to angle D in triangle DEF, then angle B in triangle ABC is congruent to
angle E in triangle DEF, and angle C in triangle ABC is congruent to angle F in
triangle DEF.
2. Corresponding sides are in proportion: When two triangles are similar, their
corresponding sides are in proportion. This means that if triangle ABC is similar to
triangle DEF, then:
AB/DE = BC/EF = AC/DF
The ratio of the lengths of any two corresponding sides is equal to the ratio of the
lengths of any other two corresponding sides.
3. The ratio of the perimeters is the same as the ratio of the corresponding sides: If two
triangles are similar, then the ratio of their perimeters is the same as the ratio of the
lengths of their corresponding sides. This means that if triangle ABC is similar to
triangle DEF, then:
(AB + BC + AC) / (DE + EF + DF) = AB/DE = BC/EF = AC/DF
4. The ratio of the areas is the square of the ratio of the corresponding sides: If two
triangles are similar, then the ratio of their areas is equal to the square of the ratio of
their corresponding sides. This means that if triangle ABC is similar to triangle DEF,
then:
Area(ABC)/Area(DEF) = (AB/DE)^2 = (BC/EF)^2 = (AC/DF)^2
5. Similarity transformations: Similar triangles can be obtained by applying similarity
transformations to a given triangle. The three types of similarity transformations are:
a. Reflection: The triangle is reflected across a line, resulting in a mirror image of the
original triangle.
b. Translation: The triangle is moved to a different position without changing its
shape or size.
