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Circles

Short Answer Type

21. If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.

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22. In figure, EF is a line passing through the centre O of a circle. If EF bisects chords AB and CD of the circle, prove that AB || CD.

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23. Two chords PQ and RS of a circle are parallel to each other and AB is the perpendicular bisector of PQ. Without using any construction, prove that AB bisects RS.

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24. OD is perpendicular to chord AB of a circle whose centre is O. If BC is a diameter, prove that CA = 2OD.

Given: OD is perpendicular to chord AB of a circle where centre is O. BC is a diameter of the circle.
To Prove: CA = 2OD
Proof: ∵ OD ⊥ AB
∴ D is the mid-point of AB
| The perpendicular drawn from the centre of a circle to a chord bisects the chord.
In ∆BAC,
∵ D is the mid-point of AB and O is the midpoint of BC
OD || AC | By mid-point theorem

$and space space space space space space space space OD space equals 1 half space AC rightwards double arrow space space space space space space space space space space CA space equals space 2 space OD$

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25. In the figure, diameter AB and a chord AC have a common end point A. If the length of AB is 20 cm and of AC is 12 cm, how far is AC from the centre of the circle?

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26. Bisector AD of ∠BAC of ∆ABC passes through the centre of the circumcircle of ∆ABC as shown in the figure. Prove that AB = AC.

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27. How will you complete an arc of a circle?

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28. Find the length of a chord which is at a distance of 3 cm from the centre of a circle whose radius is 5 cm.

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29. Two concentric circles are with centre O. A, B, C, D are the points of intersection with a line. If AD = 12 cm and BC = 8 cm, find the length of AB, CD, AC and BD.

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30. Find the length of a chord of a circle which is at a distance of 4 cm from the centre of the circle with radius 5 cm.

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