Question

In the given figure, M is the centre of the circle. Chords AB and CD are perpendicular to each other.

If ∠MAD = x and ∠BAC = y :  express ∠ABD in terms of y.

Answer 1

In the figure, M is the centre of the circle.
Chords AB and CD are perpendicular to each other at L.
∠MAD = x and ∠BAC = y

∴ Arc AD ∠AMDat the centre and ∠ABD at the remaining

(Angle in the same segment)

(Angle at the centre is double the angle at the circumference subtended by the same chord)

⇒ ∠AMD = 2∠ABD

⇒ ∠ABD = `1/2 (180° - 2x )`

⇒∠ABD = 90° - x

AB ⊥ CD, ∠ALC = 90° 

In ∆ALC,

∴ ∠LAC +∠LCA = 90°

⇒ ∠BAC + ∠DAC = 90°

⇒ y + ∠DAC = 90°

∴  ∠DAC = 90° - y

We have, ∠DAC  = ∠ABD  [Angles in the same segment]

∴ ∠ABD = 90° - y