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Circles

Short Answer Type

101.

In the figure below, ABC is a triangle in which ∠BAC = 30°. Show that BC is equal to radius of the circumcircle where centre is O.

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102. In the figure straight lines AB and CD pass through the centre O of the circle. If ∠OCE = 40° and ∠AOD = 75°, find ∠CDE and ∠OBE.

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103. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

168 Views

Long Answer Type

104.

Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

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105.
The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

Case I. When the two chords are on the same side of the centre

∵ OM 1 AB
∴ is the mid-point of AB.
| The perpendicular from the centre of a circle to a chord bisects the chord

$therefore space space space BM equals AM equals 1 half AB equals 1 half left parenthesis 6 right parenthesis equals 3 space cm$
∵ ON ⊥ CD
∴ N is the mid-point of CD.
| The perpendicular from the centre of a circle to a chord bisects the chord

$therefore space space space DN equals CN equals 1 half CD equals 1 half left parenthesis 8 right parenthesis equals 4 space cm$
In triangle OMB,

OB² = OM² + MB²
| By Pythagoras Theorem

= (4)² + (3)²
= 16 + 9 = 25
$rightwards double arrow space space space space OB equals square root of 25 equals 5 cm therefore space space space space OD space equals space OB space equals space 5 space cm space space space space space space space space space space space vertical line space Radii space of space straight a space circle$

In right triangle OND,

OD² + ON² + ND²
| By Pythagoras Theorem

$rightwards double arrow$ (5)² = ON² + ND²
$rightwards double arrow$ 25 = ON² + 16
$rightwards double arrow$ ON² = 25 - 16
$rightwards double arrow$ ON² = 9

$rightwards double arrow space space space space ON space equals square root of 9 equals space 3 space cm$
Hence, the distance of the chord from the centre is 3 cm.
Case II. When the two chords are on the opposite sides of the centre

Case I. When the two chords are on the same side of the centre
∵ OM

As in case I
ON = 3 cm.

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106.

Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

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Short Answer Type

107. Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of Us diagonals.

263 Views

108. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

571 Views

Long Answer Type

109.

AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle.

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110. Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle are