Given two equal chords AB and CD of a circle, with center O, intersecting each other at point P. Prove that : (i) AP = CP (ii) BP = DP

Draw, OM ⊥ AB and ON ⊥ CD. Join OP, OB and OD. We know that, Perpendicular to a chord, from the center of the circle, bisects the chord. ∴ OM and ON bisects AB and CD respectively. Given, Two chords are equal. ∴ AB = CD = x (let) ∴ MB = and ND = , ∴ MB = ND ..............(1) Let, MB = ND = x. From figure, OB = OD = y (Radius of same circle) In right-angled triangle OMB, By pythagoras theorem, ⇒ Hypotenuse 2 = Perpendicular 2 + Base 2 ⇒ OB 2 = OM 2 + MB 2 ⇒ OM 2 = OB 2 - MB 2 ⇒ OM 2 = y 2 - x 2 ⇒ OM = ........(2) In right-angled triangle OND, By pythagoras theorem, ⇒ Hypotenuse 2 = Perpendicular 2 + Base 2 ⇒ OD 2 = ON 2 + ND 2 ⇒ ON 2 = OD 2 - ND 2 ⇒ ON 2 = y 2 - x 2 ⇒ ON = ........(3) From equation (2) and (3), we get : ⇒ OM = ON In △ OPM and △ OPN, ⇒ ∠OMP = ∠ONP (Both equal to 90°) ⇒ OP = OP (Common side) ⇒ OM = ON (Proved above) ∴ △ OPM ≅ △ OPN (By R.H.S. axiom) We know that, Corresponding parts of congruent triangles are equal. ∴ PM = PN ..........(2) Subtracting equation (2) from (1), we get : ⇒ MB - PM = ND - PN ⇒ PB = PD ...........(4) Given, ⇒ AB = CD .........(5) Subtracting equation (4) from (5), we get : ⇒ AB - PB = CD - PD ⇒ AP = CP. Hence, proved that AP = CP and BP = DP.

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In a circle of radius 10 cm, AB and CD are two parallel chords of lengths 16 cm and 12 cm respectively. Calculate the distance between the chords, if they are on :
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In the given figure, O is the center of the circle with radius 20 cm and OD is perpendicular to AB.
If AB = 32 cm, find the length of CD.

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Mathematics

Given two equal chords AB and CD of a circle, with center O, intersecting each other at point P. Prove that :
(i) AP = CP
(ii) BP = DP

Circles

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In the given figure, O is the center of the circle with radius 20 cm and OD is perpendicular to AB.
If AB = 32 cm, find the length of CD.

Mathematics

Given two equal chords AB and CD of a circle, with center O, intersecting each other at point P. Prove that :(i) AP = CP(ii) BP = DP

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In the given figure, O is the center of the circle with radius 20 cm and OD is perpendicular to AB.If AB = 32 cm, find the length of CD.

Given two equal chords AB and CD of a circle, with center O, intersecting each other at point P. Prove that :
(i) AP = CP
(ii) BP = DP

In a circle of radius 10 cm, AB and CD are two parallel chords of lengths 16 cm and 12 cm respectively. Calculate the distance between the chords, if they are on :
(i) the same side of the center
(ii) the opposite side of the center

In the given figure, O is the center of the circle with radius 20 cm and OD is perpendicular to AB.
If AB = 32 cm, find the length of CD.