Chapter 3.1, Problem 65E

To determine

To calculate: .The length of quadrilateral of each side.

Answer to Problem 65E

The required lengths are $120\sqrt{3},160\sqrt{3},100\sqrt{3}$ and $300$ .

Explanation of Solution

Given Information:

The figure of quadrilateral which divided into two right triangle is:

  

Formula Used:

Pythagoras theorem:

  $a^2+b^2=c^2$

Where,

  $a=$ side of right angle.

  $b=$ side of right angle.

  $c=$ hypotenuse.

Calculation:

Consider the $\Delta DAB$ and $\Delta DCB$ are right angle triangle and $DB=p$ is a length of hypotenuse.

Use Pythagoras theorem in right angle $\Delta DAB$ .

  $a^2+b^2=c^2$

Substitute $6x$ for $a$ , $8x$ for $b$ and $p$ for $c$ in the above equation.

  ${\left(6x\right)}^2+{\left(8x\right)}^2=p^2$ ………………(1)

Again, use Pythagoras theorem in right angle $\Delta DCB$ .

  $a^2+b^2=c^2$

Substitute $5x$ for $a$ , $300$ for $b$ and $p$ for $c$ in the above equation.

  ${\left(5x\right)}^2+{\left(300\right)}^2=p^2$ …………….(2)

Now, compare the equation (1) and (2).

  ${\left(6x\right)}^2+{\left(8x\right)}^2={\left(5x\right)}^2+{\left(300\right)}^2$

Simplify the above equation.

  $\begin{array}{c}{\left(6x\right)}^2+{\left(8x\right)}^2={\left(5x\right)}^2+{\left(300\right)}^2\\ 36x^2+64x^2=25x^2+90000\\ 100x^2-25x^2=90000\\ 75x^2=90000\end{array}$

Simplify the equation.

  $\begin{array}{c}75x^2=90000\\ x^2=\frac{90000}{75}\\ x=\sqrt{1200}\\ x=20\sqrt{3}\end{array}$

Since, from the figure the each sides of quadrilateral are $6x,8x,5x$ and $300$ .

Substitute $20\sqrt{3}$ for $x$ in the above each side.

Thus, the side are $120\sqrt{3},160\sqrt{3},100\sqrt{3}$ and $300$ .

Hence, the required sides are $120\sqrt{3},160\sqrt{3},100\sqrt{3}$ and $300$ .