Chapter 3.2, Problem 10PSA

To determine

The value of x is to be found. Also, the congruency of the triangles $\Delta ABC\cong \Delta ADC$ is to be determined

Answer to Problem 10PSA

The value of $x=\frac{11}{2}$ , and

Yes, the congruency holds true

Explanation of Solution

Given information : The following information is given

  $\begin{array}{l}\overrightarrow{BC}=4x\\ \overrightarrow{AB}=3x+4\\ \overrightarrow{CD}=6x-11\\ \overrightarrow{DA}=5x-7\end{array}$

Firstly, the perimeter of ABCD is given, so we add all the sides to get the value of x . hence, we get

  $85=4x+3x+4+6x-11+5x-7\to x=\frac{11}{2}$

Now, putting the value of x in the equations of side given to us, we get

  $\overrightarrow{BC}\cong \overrightarrow{CD}$

  $\overrightarrow{AB}\cong \overrightarrow{AD}$

And, AC is the common chord

Thus, we can say that as all the three sides of the triangles are equal,

  $\Delta ABC\cong \Delta ADC$

Hence, the given congruency holds true