Irska

Question

AB is a chord of length 6 in a circle of radius 5 and with centre O. A square is symmetrically inscribed in the sector OAB with two vertices on the circumference. Find the area of the square.

Accepted Answer

Let XYZ be the square with Y and Z on the circle, X on OA and Y on OB. Let 2a = |YZ|. Draw OD YZ, D on YZ. Then OD AB and bisects AB. So |OA| = 5, |AC| = 3 |OC| = 4 |XE||OE| = |AC||OC| = 34 a|OE| = 34 |OE| = 4a3. Also |OD|^2 = |OY|^2 - |YD|^2 = 25 - a^2. Now |OD| = |OE| + |ED| = 4a3 + 2a = 10a3 |OD|^2 = 100a^29. So, 100a^29 = 25 - a^2 from which we get that the area of the square is equal to 4a^2 = 900109.