Chapter 10.8, Problem 18PSC

a)

To determine

To find : the value of WY

Answer to Problem 18PSC

  $WY=(12+6\sqrt{3})$ units

Explanation of Solution

Given Information :

The radius of each circle is $3$ . $\Delta WXY$ is equilateral triangle.

  

Formula Used:

Area of rectangle = Length × Breadth

Area of triangle = ½ × Base × Height

  $\because$ In $\Delta LPX$ and $\Delta MQX,\Delta LPX\cong \Delta MQX$ (Units RHS)

Calculation :

Let XL = x

Now

  $\therefore$ Area of $\Delta XWY=3\times$ (area of quadrilateral LQPV) $6$ × (area of $\Delta XQL$ ) + Area of $\Delta PQR$

  $\begin{array}{l}=\frac{\sqrt{3}}{4}{(12+2x)}^2=3\times (12\times 3)+6x\left(\frac{1}{2}\times x\times 3\right)+\frac{\sqrt{3}}{4}{(12)}^2\Rightarrow x^2+12x-3\sqrt{3}x-36\sqrt{3}=0\\ \Rightarrow (x-3\sqrt{3})(x+12)=0\\ \therefore x=3\sqrt{3}orx=-12(notvalid)\\ \therefore WY=(12+6\sqrt{3})\text{ units}\end{array}$

b)

To determine

To find : Ratio of perimeter of $\Delta ABC,\Delta PQR$ and $\Delta WXY$ .

Answer to Problem 18PSC

Perimeter ratio = $1:2(2+\sqrt{3})$

Explanation of Solution

Given Information :

Radius of circle = $3$ units, XWY is equivalent triangle

Calculation :

Ratio of perimeter $\Delta ABC:\Delta PQR:\Delta WXY$

Perimeter of $\Delta ABC=6\times 3=18$ units

Perimeter of $\Delta PQR=12\times 3=36$ units

Perimeter of $\Delta WXY=3\times (12+6\sqrt{3})$ units

  $\therefore$ Perimeter of = $18:36:3\times (12+6\sqrt{3})=1:2:(2+\sqrt{3})$