The length of arc APD is: [tex]\frac{\pi r}{2}[/tex]
A square when inscribed in a circle will fit the circle such that, the 4 edges of the square touches the sides of the circle. The radius of the circle can be drawn from any of the 4 edges.
Given that ABCD is a square:
This means that:
[tex]AB = BC = CD = DA[/tex] --- equal side lengths
To calculate the length of arc APD, we make use of the following arc length formula
[tex]APD = \frac{\theta}{360} * 2\pi r[/tex]
Where
[tex]\theta = \angle ADO[/tex] and O is circle center
Since ABCD is a square, then:
[tex]\theta = \angle ADO = 90^o[/tex]
So, we have:
[tex]APD = \frac{90}{360} * 2\pi r[/tex]
[tex]APD = \frac{1}{4} * 2\pi r[/tex]
[tex]APD = \frac{\pi r}{2}[/tex]
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