Four horses are tethered at four corners of a square plot of side $63\text{ metres}$ so that they just cannot reach one another. The area left ungrazed is

### Concept & Area of Grazing Sectors This is practically identical to the "four touching circles" problem. If the horses "just cannot reach one another", their grazing regions represent four quarter-circles that meet exactly at the midpoints of the square's sides. $$\text{Ungrazed Area} = \text{Total Plot Area} - \text{Total Grazed Area}$$ ### Step-by-Step Solution * **Given:** Side of the square plot $s = 63\text{ m}$. * **Deduction:** The radius $r$ of each horse's grazing sector is exactly half the side length: $r = \frac{63}{2} = 31.5\text{ m}$. * **Calculation:** Area of the square plot = $63^2 = 3969\text{ m}^2$. * **Calculation:** The four $90^\circ$ grazing sectors form one full circle of radius $31.5\text{ m}$. Area grazed = $\pi r^2$. * **Calculation:** $\text{Grazed Area} = \frac{22}{7} \times \frac{63}{2} \times \frac{63}{2} = 11 \times 9 \times \frac{63}{2} = 99 \times 31.5 = 3118.5\text{ m}^2$. * **Calculation:** Ungrazed Area = $3969 - 3118.5 = 850.5\text{ m}^2$. ### Exam Strategy & Shortcut Using the derived formula from the four touching circles concept, the ungrazed area is directly $\frac{6}{7}r^2$. $\frac{6}{7} \times (\frac{63}{2})^2 = \frac{6}{7} \times \frac{3969}{4} = \frac{3}{7} \times \frac{3969}{2} = 3 \times \frac{567}{2} = \frac{1701}{2} = 850.5$. This significantly reduces the arithmetic steps. ### Common Pitfall A common misstep is calculating the area of four separate circles instead of realizing the grazing areas inside the plot only amount to four quarter-sectors (one single full circle equivalent). ### Final Answer Therefore, the correct answer is **$850.5\text{ m}^2$**.