Question

In the square below, the diagonal AC is 12v2 inches Find the area of the shaded region and find the exact circumference of the inscribed © X.

Answer 1

`AC=12\sqrt[]{2}`

AB = BC

`\begin{gathered} (AC)^2=(AB)^2+(BC)^2 \\ (12\sqrt[]{2})^2=(BC)^2+(BC)^2 \end{gathered}`
`\begin{gathered} 24=2(BC)^2 \\ \frac{24}{2^{}}=(BC)^2\text{ } \\ (BC)^2=12 \\ BC\text{ =}\sqrt[]{12} \\ BC\text{ = 2}\sqrt[]{3}\text{ inches} \end{gathered}`

Area of shaded part = Area of the square - the area of the circle

`\begin{gathered} \text{Area of square = length x length} \\ \text{Area of square = 2}\sqrt[]{3}*2\sqrt[]{3}=12inch^2 \end{gathered}`
`\begin{gathered} \text{Area of circle = }\pi\text{ }* r^2 \\ r=\text{ BC = 2}\sqrt[]{3}inches \\ \text{Area of circle= 3.14 }*(2\sqrt[]{3)}^2=\text{ 37.68} \end{gathered}`

Area of shaded part = 37.68- 12 =25.68 square inche

`\text{Circumference of a circle = 2}*\pi* r`