Chapter 5.4, Problem 16PSB

a.

To determine

To find: The values of $x\text{ and }y$ .

Answer to Problem 16PSB

  $x=5,y=7$ .

Explanation of Solution

Given information:

A figure is given as below.

  

Calculation:

Given $AB=x+3,BC=x+4,CD=2x-1,\text{ and }AD=3x-y$ .

  

In above figure,

  $\begin{array}{l}CD=BC\\ \Rightarrow 2x-1=x+4\\ \Rightarrow x=5\\ \text{Also, }AD=AB\\ \Rightarrow 3x-y=x+3\\ \Rightarrow 2x-y=3\\ \Rightarrow 2\left(5\right)-y=3\\ \Rightarrow y=10-3\\ \Rightarrow y=7\end{array}$

Hence, value are $x=5,y=7$ .

b.

To determine

To find: The perimeter of the kite.

Answer to Problem 16PSB

  $34$ units.

Explanation of Solution

Given information:

Same as part $a$ of this exercise.

Calculation:

Given $AB=x+3,BC=x+4,CD=2x-1,\text{ and }AD=3x-y$ .

  

From part $a$ of this exercise $x=5,y=7$ .

  $\begin{array}{l}\text{So, perimeter of the kite}=AB+BC+CD+AD\\ \Rightarrow \text{Perimeter of the kite}=\left(x+3\right)+\left(x+4\right)+\left(2x-1\right)+\left(3x-y\right)\\ \Rightarrow \text{Perimeter of the kite}=7x-y+6\\ \Rightarrow \text{Perimeter of the kite}=7\left(5\right)-7+6\\ \Rightarrow \text{Perimeter of the kite}=34\end{array}$

Hence, perimeter of the kite is $34$ units.

c.

To determine

To find: If it is possible for $AC$ to be $19$ units.

Answer to Problem 16PSB

Length of $AC$ cannot be $19$ units.

Explanation of Solution

Given information:

Same as part $a$ of this exercise.

Calculation:

Given $AB=x+3,BC=x+4,CD=2x-1,\text{ and }AD=3x-y$ .

  

From part $a$ of this exercise $x=5,y=7$ .

  $\begin{array}{l}CD=2\left(5\right)-1=9\\ AD=3\left(5\right)-7=8\end{array}$

For a triangle, sum of any two side is greater than third side.

In case of $\Delta ACD$ , $AC\text{ should be less than }\left(AD+CD=9+8=17\right)$ .

Hence, length of $AC$ cannot be $19$ units.