Let $X, Y, Z$ be respectively the areas of a regular pentagon, regular hexagon and regular heptagon which are inscribed in a circle of radius 1 . Then

• A

  $\frac{\mathrm{x}}{5} < \frac{\mathrm{Y}}{6} < \frac{\mathrm{Z}}{7}$ and $\mathrm{X} < \mathrm{Y} < \mathrm{Z}$
• B

  $\frac{\mathrm{x}}{5} < \frac{\mathrm{Y}}{6} < \frac{\mathrm{Z}}{7}$ and $\mathrm{X}>\mathrm{Y}>\mathrm{Z}$
• C

  $\frac{\mathrm{x}}{5}>\frac{\mathrm{Y}}{6}>\frac{\mathrm{Z}}{7}$ and $\mathrm{X}>\mathrm{Y}>\mathrm{Z}$,
• Correct answer
  D

  $\frac{\mathrm{x}}{5}>\frac{\mathrm{Y}}{6}>\frac{\mathrm{Z}}{7}$ and $\mathrm{X} < \mathrm{Y} < \mathrm{Z}$

Solution

Area of Pentagon $=5 \times \frac{1}{2} \sin 72^{\circ}=2.377=\mathrm{x}$ Area of hexagon $=6 \times \frac{1}{2} \sin 60^{\circ}=3 \times \frac{\sqrt{3}}{2}=2.598=\mathrm{y}$ Area of heptagon $=7 \times \frac{1}{2} \sin \frac{360^{\circ}}{7}=2.73=\mathrm{z}$

Hence, the correct answer is option: 'D'.