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 :

1. express ∠AMD in terms of x.
2. express ∠ABD in terms of y.
3. prove that : x = 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

i. In ∆AMD,

MA = MD

∴ ∠MAD = ∠MDA = x

But in ∆AMD,

∠MAD + ∠MDA + ∠AMD = 180°

`=>` x + x + ∠AMD = 180°

`=>` 2x + ∠AMD = 180°

`=>` ∠AMD = 180° – 2x

ii. ∴ Arc AD∠AMD at 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^circ - 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

iii. We have, ∠ABD = 90° – y and ∠ABD = 90° – x  [Proved]

∴ 90° – x = 90° – y

`=>` x = y