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 :
(i) express ∠AMD in terms of x.
(ii) express ∠ABD in terms of y.
(iii) prove that : x = y.

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 :
(i) express ∠AMD in terms of x.
(ii) express ∠ABD in terms of y.
(iii) prove that : x = y.

KnowledgeBoat · Accepted Answer

Mark the point of intersection of AB and CD as L. (i) In △AMD, MA = MD [Radius of circle] ∴ ∠MAD = ∠MDA = x. In △AMD, ⇒ ∠MAD + ∠MDA + ∠AMD = 180° [Angle sum property of triangle] ⇒ x + x + ∠AMD = 180° ⇒ ∠AMD = 180° - 2x. Hence, ∠AMD = 180° - 2x. (ii) Let the perpendicular chords AB and CD intersect each other at L. From figure, ∠ALC = 90°. In △ALC, ⇒ ∠LAC + ∠LCA + ∠ALC = 180° [Angle sum property of triangle] ⇒ y + ∠DCA + 90° = 180° ⇒ ∠DCA = 180° - 90° - y ⇒ ∠DCA = 90° - y From figure, ∠ABD = ∠DCA [Angles in same segment are equal] ∴ ∠ABD = 90° - y. Hence, ∠ABD = 90° - y. (iii) We know that, Angle which an arc subtends at the center is double that which it subtends at any point on the remaining part of the circumference. ∴ ∠AMD = 2∠ABD ∠ABD = ∠AMD = = 90° - x. We have, ∠ABD = 90° - y and ∠ABD = 90° - x ⇒ 90° - y = 90° - x ⇒ x = y. Hence, proved that x = y.

Mathematics

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 :
(i) express ∠AMD in terms of x.
(ii) express ∠ABD in terms of y.
(iii) prove that : x = y.

Circles

Related Questions

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Mathematics

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 :(i) express ∠AMD in terms of x.(ii) express ∠ABD in terms of y.(iii) prove that : x = y.

Circles

Related Questions

In the given figure, PQ is the diameter of the circle whose center is O. Given, ∠ROS = 42°, calculate ∠RTS.

In the given figure, AOB is a diameter and DC is parallel to AB. If ∠CAB = x°; find (in terms of x) the values of :(i) ∠COB,(ii) ∠DOC,(iii) ∠DAC,(iv) ∠ADC.

In the given figure, AC is the diameter of circle, centre O. Chord BD is perpendicular to AC. Write down the angles p, q and r in terms of x.

The given figure shows a circle with center O and ∠ABP = 42°. Calculate the measure of :(i) ∠PQB(ii) ∠QPB + ∠PBQ

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 :
(i) express ∠AMD in terms of x.
(ii) express ∠ABD in terms of y.
(iii) prove that : x = y.

In the given figure, AOB is a diameter and DC is parallel to AB. If ∠CAB = x°; find (in terms of x) the values of :
(i) ∠COB,
(ii) ∠DOC,
(iii) ∠DAC,
(iv) ∠ADC.

The given figure shows a circle with center O and ∠ABP = 42°. Calculate the measure of :
(i) ∠PQB
(ii) ∠QPB + ∠PBQ