Short Answer TypeIn 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.
Long Answer TypeTwo 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.
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?
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.
Short Answer Type
To Prove: The circle drawn with any sides AB of rhombus AB DC as a diameter passes through the point E.
Proof: In ∆AEB and ∆AEC,
AB = AC. | Given
∠BEA + ∠CEA = 180° | Linear Pair
⇒ 90° + ∠CEA = 180°
| Angle in a semi-circle
⇒ ∠CEA = 90°
∴ ∠BEA = ∠CEA = 90°
AE = AE | Common
∴ ∆AEB ≅ ∆AEC | R.H.S.
∴ BE = CE | C.P.C.T.
⇒ E is the mid-point of BC
⇒ E is the point of intersection of diagonals
Hence, the circle drawn with AB as diameter passes through the point E.
Similarly, we can prove that the circle drawn with AC as diameter passes through the point D which is the point of intersection of its diagonals.
Long Answer TypeAC 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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