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प्रश्न
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.
उत्तर
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
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