Question

Question #34468

Answer 1

Case 1 : Circumcircle `R_c = color(green)(a / sqrt2`

Case 2 : Incircle `R_i = color(blue)( a/2`

Explanation:

Case 1 : Circumcircle

Let a be the side of the square ABCD.

O is the intersection point of diagonals AC & BD.

Diagonals bisect each other 90 degrees (Properties of a square

`:. OA = OB = OC = OD = R_c` where `R_c` is the radius of the circumcircle

Diagonals also bisect the angles at the vertices.

Therefore, `OhatDC = OhatCD = (AhatDC)/2 = (BhatCD)/2 = 45^0`

Triangle ODC is isosceles with base angles 45 degrees and the sides are in the ratio,

`DC : OC : OD = a : a/sqrt2 : a/sqrt2`

Hence radius of the circumcircle of square ABCD is

`R_c = color(green)(a / sqrt2`

Case 2 : Incircle

It is a straight forward case.

O is the center where the diagonals will intersect each other at right angles.

Radius of incenter `R_i = OE = OF =color(blue)( a/2`