Chapter 8, Problem 25RP

To determine

To calculate: The perimeter of the given triangle

Answer to Problem 25RP

The perimeter of the given triangle is 30 cm

Explanation of Solution

Given information:

One side of the triangle is 4 cm shorter than a second side

The ray bisecting the angle formed by these sides divides the opposite side into 4 cm and 6 cm segments

Formula used:

Perimeter is sum of sides

Ratio of corresponding sides of the triangles are equivalent

Calculation:

  

In the figure,

Let BC be x cm then AB will be ( x − 4) cm

Ray BD divides $\angle ABC$ such that AD = 4 cm and CD = 6 cm

Triangles ABD and DBC are similar because

  $\angle \text{ADB=}\angle \text{BDC}$ , since they are opposite to common arm

  $\Rightarrow$

  $\angle ADB=\angle BDC$

  $=\angle ABC$

AAA criteria of similarity is satisfied

Therefore, ratio of corresponding sides will be equivalent

Hence,

  $\begin{array}{l}\frac{AB}{BC}=\frac{AD}{DC}\\ \Rightarrow \frac{x-4}{x}=\frac{4}{6}\\ \Rightarrow 6\left(x-4\right)=4x\\ \Rightarrow 6x-4x=24\\ \Rightarrow 2x=24\\ \Rightarrow x=12\end{array}$

And perimeter of triangle

  $\begin{array}{l}=x+x-4+4+6\\ =2x+6\end{array}$

At x = 12,

Perimeter = 2.12+6=30 units