# Geometry Math Riddle: Two Circles are Inscribed inside a Square

## Solve the geometry math riddle two circles are inscribed inside a square

The figure shows two circles are inscribed inside a square and when OA and OB are chords of the circles, What is the angle between the chords (∠AOB =?)

## Solution

Connect the diagonal of the square, draw the diameter of the bigger circle through B and connect the other end of the diameter to O

From the above figure, ΔOBC is a right angle (because BC is the diameter of the circle)

From ΔMBQ

∠MBQ = 90°

⇒ ∠BMQ = ∠BQM = (180 – 90)/2

⇒ ∠BMQ = ∠BQM = 45°

From ΔOBM

∠OMB = 180 – 45 = 135°

⇒ ∠MOB = ∠OBM = (180 – 135)/2 {MO = MB}

⇒ ∠MOB = ∠OBM = 22.5°

Now, draw the diameter of the smaller circle through A and connect the other end of the diameter to O

From the above figure, ΔAOC is a right angle (because AC is the diameter of the circle)

From ΔPAN

∠PAN = 90°

⇒ ∠APN = ∠ANP = (180 – 90)/2

⇒ ∠APN = ∠ANP = 45°

From ΔOAN

∠ONA = 180 – 45 = 135°

⇒ ∠AON = ∠OAN = (180 – 135)/2 {AN = ON}

⇒ ∠AON = ∠OAN = 22.5°

From figure

∠AOB = 180 – (∠AON + ∠BOQ)

⇒ ∠AOB = 180 – (22.5 + 22.5)

⇒ ∠AOB = 135°

That is **angle between the chords is 135°**