The aim of this question is to find the **length** of a **side** of a **triangle** that is **similar** to another **triangle** whose **lengths** of the **sides** are given.

This question depends on the concepts of **triangle geometry. Two triangles** are said to be **similar** if one of the following criteria is met between those **two triangles.**

**Two pairs of the corresponding angles of both triangles are equal.****Three sides of both triangles are equal.****Two corresponding sides of both triangles are proportional to each other, and the angle between them is equal.**

## Expert Answer

Figure 1 shows the **diagram** of the triangles.

The given information for this question is given as follows:

\[ \triangle ABC\ and\ \triangle ADE\ are\ two\ triangles \]

\[ AD = 20 \]

\[ DB = 25 \]

\[ DE = 15 \]

\[ BC = X \]

The side **BC** needs to be determined using **similar triangles.**

Observing the **diagram,** we can evaluate that the corresponding **sides DE** and **BC** are **proportional** to each other.

\[ DE = k\ BC \]

There is a **common angle** between **two triangles,** which is given as:

\[ \angle BAC\ and\ \angle DAE\ are\ COMMON\ Angles \]

There are **two angles** that are **congruent** to each other as they have a **common line** and other **lines** are **parallel.** The **angles** are given as:

\[ \angle ADE \cong \angle ABC \]

Using the **similar triangles rule** of having **two pairs** of **angles** of **two triangles** as **equal,** the triangles will be **similar** to each other. We can use the **proportional relationship** of both **triangles** to calculate the **side length** as:

\[ \dfrac{ AB }{ AD } = \dfrac{ BC }{ DE } \]

Substituting the values, we get:

\[ \dfrac{ 45 }{ 20 } = \dfrac{ X }{ 15 } \]

Rearranging the equation for **X,** we get:

\[ X = \dfrac{ 15 \times 45 }{ 20 } \]

\[ X = \dfrac{ 675 }{ 20 } \]

\[ X = 33.75 \]

## Numerical Result

The **side length** of the **triangle** is calculated to be:

\[ X = 33.75 \]

## Example

Find the **missing length** marked as **X** of the **triangle** given below.

The given information about this question is given as follows:

\[ \triangle ABC\ and\ \triangle ADE\ are\ two\ triangles \]

\[ AD = 25 \]

\[ DB = 30 \]

\[ DE = 16 \]

\[ BC = X \]

The **triangles** have **two pairs** of **corresponding angles equal** to each other. So, we can apply the **similar triangles rule** to calculate the **side** of the **triangle.**

\[ \angle BAC = \angle DAE \rightarrow Common\ Angles \]

\[ \angle ADE \cong \angle ABC \rightarrow Congruent\ Angles \]

Using the **similar triangles rule** to calculate the **side length** using the corresponding sides as **proportionally equal** to each other.

\[ \dfrac{ AB }{ AD } = \dfrac{ BC }{ DE } \]

Substituting the values, we get:

\[ \dfrac{ 55 }{ 25 } = \dfrac{ X }{ 16 } \]

Rearranging the equation for **X**, we get:

\[ X = \dfrac{ 16 \times 55 }{ 25 } \]

\[ X = \dfrac{ 880 }{ 25 } \]

\[ X = 35.2 \]

*Images/Mathematical Drawings are created with Geogebra.*