Please answer me question#12 page#97 from following link https://tstuition.wikispaces.com/file/view/3+-+Mensuration.pdf many many thanks If...

Well, to calculate the area of the star it is more easier
to calculate the area of the pentagon and to subtract the areas of the small isosceles
triangles formed outside the star shape.

Since the pentagon
is regular, the base angles of the isosceles triangles measure 36 degrees each. The
bases of these isosceles triangles are the sides of pentagon and the measure 10
cm.

To determine area of each isosceles triangles, we'll
have to calculate its heights. We'll determine the height from any right angle triangle
formed when we draw the height within each isosceles triangles using tangent
function.

tan 36 = height (opp.)/(10/2)
(adj.)

height = 5*tan
36

height = 3.63 cm

We'll
calculate the area of isosceles triangle;

A =
base*height/2

A = 10*3.63/2

A
= 18.15 ` `

Now, we'll calculate the total area of these
isosceles triangles:

A total = 5*
18.15

A total = 90.75
`cm^(2)`

The area of pentagon is formed from the areas of
the three triangles fromed when drawing 2 diagonals of
pentagon.

To determine th length of diagonal, we'll
consider the isosceles triangle, whose equal sides are the sides of pentagon. The top
angle is one of the interior angles of pentagon and it measures 108
degrees.

Since we also need to calculate the height of
these triangles, we'll draw the height and we'll notice that inside isosceles triangle
there are formed two right angle triangles, whose hypotenuses are the sides of
pentagon.

We'll use the sine function to determine the
height and we'll use cosine fuction to determine the half length of
diagonal.

sin 36 =
height/10

height = 10*sin
36

height = 5.87

cos 36 =
(diagonal/2)/10

diagonal/2 = 10*cos
36

diagonal/2 = 8.09

diagonal
= 16.18 cm.

Area of the isosceles triangle
is:

A triangle =
5.87*16.18/2

A triangle = 47.48
`cm^(2)`

We'll determine the area of the middle isosceles
triangle.

A mid. triangle = diagonal*diagonal*sin
36/2

A mid. triangle =
`16.18^(2)`

A mid. triangle = 76.93
`cm^(2)`

The area of pentagon = 2*A triangle + A mid.
triangle

Area pentagon = 2*47.48 +
76.93

Area pentagon = 171.89
`cm^(2)`

The area of the star = A pentagon - A
total

The area of the star = 171.89 -
90.75

The area of the star = 81.14 `cm^(2)`