**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.*

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