Equal Chords

Q.

State true or false for the following statement:

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection, then the chords are equal.

1. True

2. False

Q.

In a figure, If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D. Prove that AB = CD

Q.

AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Then AE = CE.

1. True

2. False

Q. If arcs AXB and CYD of a circle are congruent, then what is the ratio of chord AB and chord CD?

1. 2 : 1
2. 1 : 1
3. 1 : 2
4. None of these

Q.

Two chords intersect at a point within the circle and the diameter through this point bisects the angle between the chords. Then

1. one of the chords is twice the length of the other.

2. one of the chords is four times the length of the other.

3. two chords have the same length.

4. one of the chords is one third the length of the other.

Q.

In the figure, O is the center of the circle of radius 5 cm. P and Q are points on chords AB and CD such that OP $\perp$ AB, OQ $\perp$ CD, AB || CD, AB = 6 cm and CD = 8 cm. Determine PQ.

1. 7 cm

2. 8 cm

3. 5 cm

4. 6 cm

Q.

If each of two chords, $AB$ and $CD$, of a circle are at a distance of 4 cm from the centre, then $AB=CD$.

1. True

2. False

Q.

AB and CD are two equal chords of a circle with centre O which intersect each other at right angles at point P. If the perpendiculars from the centre to AB and CD meet them at M and N respectively, then MONP is a_______.

1. square

2. trapezium

3. kite

4. rhombus

Q.

The angle made by two equal chords drawn from a point on the circle is bisected by the diameter through that point.

1. False

2. True

Q.

AB and CD are two equal chords of a circle. If chord AB is at a distance of 4 cm from the centre, then the distance of chord CD from the centre is

1. 4 cm

2. 6 cm

3. 2 cm

4. 3 cm

Q.

If each of two chords, AB and CD of a circle are at a distance of 4 cm from the center, then AB = CD.

1. True

2. False

Q.

Question 10
If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

Q. If AB, BC and CD are equal chords of a circle with O as centre and AD diameter, than ∠AOB =

(a) 60°

(b) 90°

(c) 120°

(d) none of these

Q.

In a circle of radius 17 cm, two parallel chords of lengths 30 cm and 16 cm are drawn.

Prove that:
(i) distance between the chords, if both the chords are on the opposite sides of the centre, is 23 cm, and
(ii) distance between the chords, if both the chords are on the same side of the centre is 7 cm.

Q. Two chords AB and CB of a circle are at equal distances from the centre of the circle. Then, $△$ABC is an isosceles triangle.

1. True
2. False

Q.

In the following figure, if chords AB and CD are equal to 6 cm, then the distances of the chords from the centre of the circle of radius 5 cm are ___ and ___ respectively.

Q.

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Q.

Two chords AB and CD of a circle are each at a distance of 5 cm from centre. Then AB=CD

1. False

2. True

Q. The following figure shows a circle with centre O. If OP is perpendicular to AB, then AP$=$BP.

1. True
2. False

Q.

AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Then EB = ED.

1. True

2. False

Q. It two equal chords of a circle intersect within the circle. Prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Q. In any triangle ABC, if the angle bisector of $\angle A$ and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC.

Q. In the given figures, O is the centre of the circles. Find the respective values of x.

1. 60°
2. 115°
3. 40°

Q.

In the given figure, it is given that chords AB = PQ. The $\angle AOB={55}^0$, and $\angle POQ=x^0$, find $x$ (in degrees).

1. 55

2. 25
3. 35
4. 45

Q. In the given figures, O is the centre of the given circles. Match the values of $x$.

1. ${40}^{\circ }$
2. ${60}^{\circ }$
3. ${200}^{\circ }$

Q. PQ and RQ are the chords of a circle equidistant from the centre. Prove that the diameter passing through Q bisects ∠PQR and ∠PSR.

Q. If two circles intersect at two distinct points, prove that the line through their centres is the perpendicular bisector of their common chord.

Q.

AB and CD are two chords of a circle such that both are at a distance of 5 cm from the centre.If CD = 3 cm, then AB =

1. 1 cm

2. 3 cm

3. 5 cm

4. 7 cm

Q.

AB and CD are two chords of a circle such that AB = CD = 3.9 cm. If chord AB is at a distance of 2.6 cm from the centre, then the distance of chord CD from the centre is

1. 0.6 cm

2. 1.6 cm

3. 2.6 cm

4. 3.6 cm

Q. In any triangle ABC, if the angle bisector of $\angle A$ and perpendicular bisector of BC intersect, prove that they intersect on the circum circle of the triangle ABC.