The square ABCD has an area of 81 cm² - Leaving Cert Mathematics - Question Question 1 - 2014

To find the area of the sector with radius 9 cm, we use the formula for the area of a sector:

$A = \frac{1}{4} \times \pi r^2$

Substituting the radius:

$A = \frac{1}{4} \times \pi \times (9)^2 = 63.6 \, \text{cm}^2.$ Therefore, the area of the sector is approximately 63.6 cm².

The area of the shaded region can be calculated by subtracting the area of the first sector from the area of the square:

$81 - 63.6 = 17.4 \, \text{cm}^2.$ Therefore, the area of the shaded region is approximately 17.4 cm².

Given that |P| is on the arc, the triangle APC is isosceles. The area can be found using the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times base \times height.$
Assuming |AC| is the base and calculating the height using the coordinates, we find:

$APC = \frac{1}{2}(9)(\sqrt{\frac{18 - \sqrt{2}}{2}})= 16.8 \, \text{cm}^2.$
Thus, the area of triangle APC is approximately 16.8 cm².