Similar Triangles Formula

Before we learn similar triangles formula we must understand when are two figures said to be similar. If one figure can be obtained from another by a sequence of transformations such as resizing, flipping, sliding, or turning. That is, similar figures have the same shape but not necessarily the same size.

Two triangles are said to be similar if their:

• corresponding angles are equal
• corresponding sides are in the same ratio

However, to ensure that the two triangles are similar, we do not necessarily need to have information about all sides and all angles. Let us learn about the similar triangles formula.

What Is Similar Triangles Formula?

There are three criteria to determine if two triangles are similar.

1. AA (Angle Angle): If any two of the angles of the triangles are equal, then the triangles are said to be similar.
2. SAS (Side Angle Side): If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar.
3. SSS (Side Side Side): If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

The symbol used to denote the similarity between triangles is '~'.
Triangles \(ABC\) and \(DEF\) are similar is denoted by \(\bigtriangleup ABC\sim \bigtriangleup DEF\).

Let us see the applications of the similar triangles formula in the following section.

Examples Using the Similar Triangles Formula

Example 1: The dimensions of triangles ABC and DEF are as follows:

AB = 4 units, BC = 5 units, AC = 6 units

DE=16 units, EF=20 units, DF=24 units

Using similar triangles formula check if the triangles are similar.

Solution:

Determine the ratio of the corresponding sides of the triangles to check if they are similar.

Take the ratio of the shortest sides of both the triangles and the ratio of the longest sides of both the triangles.

\(\begin{align}\dfrac{AB}{DE}&=\dfrac{4}{16}=\dfrac{1}{4}\\\dfrac{BC}{EF}&=\dfrac{5}{20}=\dfrac{1}{4}\\\dfrac{AC}{FG}&=\dfrac{6}{24}=\dfrac{1}{4} \end{align}\)

Since the corresponding sides of the triangles are in the same ratio, therefore they are similar.

Answer: \(\bigtriangleup ABC\sim \bigtriangleup DEF\)

Example 2: Is every pair of equilateral triangles similar?

Solution:

The measure of each angle in an equilateral triangle is \(60^{\circ}\).

Since the corresponding angles in every pair of the equilateral triangle are equal to \(60^{\circ}\), the triangles are similar.

Answer: Yes, every pair of equilateral triangles is similar.

Example 3: A pole of height 2 yards casts a shadow of length 4 yards. A tree casts a shadow of 24 yards.

Find the height of the tree, if it is known that the triangles formed by joining the tip of the tree and the shadow of the tree, are similar to the triangle formed by joining the tip of the pole with the tip of the shadow of the pole.

Solution:

Here, we can see that △PQR is similar to △ABC

Since the corresponding sides of similar triangles are in the same ratio,

we get, PQ/AB = QR/BC = PR/AC

Putting the values, we get

PQ/2 = 24/4

On solving we get,

PQ = 12

Answer: The height of the tree is 12 yards.