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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.

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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?

Given: Diameter AB and a chord AC have a common end point A. AB = 20 cm and AC = 12 cm. To determine: OD
Determination: ∵ OD ⊥ AC

$therefore space space space space AD equals DC equals 1 half AC equals 1 half cross times 12 equals 6 space cm$

| ∵ The perpendicular drawn from the centre of a circle to a chord bisects the chord.

$OA equals OB equals 1 half AB equals 1 half cross times 20 equals 10 space cm$
In right triangle ODA,
OA² = OD² + AD²| By Pythagoras Theorem
⇒ (10)² = OD² + (6)²⇒ OD = 8 cm
Hence, AC is 8 cm far 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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