Chapter 11.6, Problem 27WE

(a)

To determine

Draw the figure using only compass.

Explanation of Solution

Given :

  

Calculation:

Start by making a circle with radius 6 units.

  

Now , select any point on the circumference of the circle and make an arc with radius 6 units.

  

  

Now, from the point where the arc intersects the circle, make another arc with radius 6 units.

  

  

Continue this way:

  

  

  

  

Hence , we get the given figure only via compass.

(b)

To determine

Find the area of the 6 petals.

Answer to Problem 27WE

39.12 square units.

Explanation of Solution

Given :

Radius of the circle is 6 units .

  

Calculation:

All the 6 petals are congruent.

We find area of 1 petal first , and then multiply it by 6 .

  

It forms 6 congruent triangles , the central measure of each triangle is

  $\frac{360°}{6}=60°$

Also , the other two sides of triangle are equal .

So, the triangle is equilateral.

And the three petals are on the three sides of the triangle.

We can find the area of half of the petal by subtracting the area of equilateral triangle FAE from the area of sector FAE :

  $\begin{array}{l}\frac{1}{6}\pi r^2-\frac{\sqrt{3}}{4}{(side)}^2\\ =\frac{1}{6}\pi 6^2-\frac{\sqrt{3}}{4}{6}^2\\ \approx 18.85-15.59\\ =3.26\end{array}$

So, the area of 1 petal is $2\times 3.26=6.52$ .

Now, the area of 6 petals or the area of the shaded region = $6\times 6.52=39.12$ square units.