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Q.

Figure shows a regular pentagon with all its vertices on the sides of a rectangle . If BC = $\frac{1}{4}$ , calculate the perimeter of the pentagon .

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a

0.31

b

0.62

c

0.77

d

1.00

answer is C.

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Detailed Solution

Concept- Using the sine rule

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}

, when a , b and c are the length of sides of the triangle and A, B,and C are the angles opposite to that sides a,b and c respectively .assume the length of one the sides of pentagon as x. we will use the sine rule to determine the value of x and we will multiply this value with 5 to find the perimeter.
All sides of the regular pentagon are equal. Assuming each side as x. name the pentagon MNOPQ. Now join the vertices N and Q to form a triangle . MNQ . also there , a rectangle BNQC formed.
C:UsersuserAppDataLocalMicrosoftWindowsINetCacheContent.Worda46cc23b-3ccf-446c-bc2c-0f0cc7d3c3151661398243084875650.jpeg

C:UsersuserAppDataLocalMicrosoftWindowsINetCacheContent.Worda46cc23b-3ccf-446c-bc2c-0f0cc7d3c3151661398243084875650.jpeg

NQ = BC = \frac{1}{4}

since they are the opposite sides of rectangle BNQC
In triangle MNQ
Angle NMQ=108°, all internal angles of a regular pentagon is 108°.

MN = MQ,

angle MNQ = Angle MQN = 36 °

Apply sine rule in triangle in MNQ

\frac{MN}{\sin 36 °} = \frac{NQ}{\sin 108 °}

NQ = BC = \frac{1}{4} we get

\frac{MN}{\sin 36 °} = \frac{NQ}{\sin 108 °}

\frac{x}{\sin 36 °} = \frac{1}{4 \sin 108 °}

Using the conclusion,

\sin 108 ° = \sin (180 ° - 108 °) = \sin 72 ° and

\sin 2 x = 2 sin xcos x, we get

x = \frac{\sin 36 °}{4 \times 2 \sin 36 ° \cos 36 °}

x = \frac{1}{8 \times \cos 36 °}

Substituting , Cos 36

° = \frac{\sqrt{5 + 1}}{4}, we get

x = \frac{1}{8 \times \frac{\sqrt{5 + 1}}{4}}

x = \frac{1}{2} \times \frac{1}{\sqrt{5 + 1}}

Rationalise denominator.

x = \frac{1}{2} \times \frac{1}{\sqrt{5 +}} \times \frac{\sqrt{5 - 1}}{\sqrt{5 - 1}}

= \frac{1}{2} \times \frac{\sqrt{5 - 1}}{5 - 1}

= \frac{\sqrt{5 - 1}}{8}

\sqrt{5} = 2.236

x = \frac{2.236 - 1}{8}

x = \frac{1.236}{8}

x = 0.1545

Perimeter of the pentagon =5x because all 5 sides are equal.

perimetre = 5 \times 0.1545 = 0.77

Hence, option (3) is the correct answer.

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