SSS, SAS, AAS, ASA, RHS Criteria for Congruency of Triangles

Q.

Question 3
If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

Q.

Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

Q.

Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

Q.

The adjoining diagram shows a Pentagon inscribed in a circle, center $O.$ Given $AB=BC=CD$ and $\angle ABC={132}^{\circ }$. Calculate the value of

(i) $\angle AEB$

(ii) $\angle AED$

(iii) $\angle COD$

Q.

In the given figure, BOC is a diameter of a circle with centre O. If AB and CD are two chords such that AB||CD and AB = 10 cm then CD = ?

(a) 5 cm

(b) 12.5 cm

(c) 15 cm

(d) 10 cm

Q.

In the adjoining figure, O is the centre of a circle. If AB and AC are chords of the circle such that AB = AC, OP $\perp$AB and OQ $\perp$ AC, prove that PB = QC.

Q. Question 2
If the perpendicular bisector of a chord AB of a circle PXAQBY intersect the circle at P and Q, prove that arc PXA $\cong$ arc PYB.

Q.

In the figure, O is the centre of the circle, BO is the bisector of $\angle ABC$. Show that AB = BC.

Q.

In the adjoining figure, AB and AC are two equal chords of a circle with centre O. Show that O lies on the bisector of $\angle$BAC.

Q.

In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD, prove that AB = CD.

Q.

Question 2
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Q.

In the adjoining figure, OPQR is a square. A circle drawn with centre O cuts the square in X and Y. Prove that QX = QY.

Q.

In the construction of the bisector of a given angle, as shown in the figure below

ΔBEF $\cong$ ΔBDF by which congruence criterion?

1. AAS

2. SSS

3. RHS

4. SAS

Q. AB and CD are two parallel chords of a circle with centre O such that AB = 6 cm and CD = 12 cm. The chords are on the same side of the centre and the distance between them is 3 cm. The radius of the circle is

(a) 6 cm

(b) $5\sqrt{2}\mathrm{cm}$

(c) 7 cm

(d) $3\sqrt{5}\mathrm{cm}$

Q. Q4) Two chords AB, CD of lengths 5 and 11 cm respectively of a circle are parallel if the distance between A and CD is 3 cm, find the radius of the circle

Q. Question 3
If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

Q. Two circles of radii 10 cm and 8 cm intersect each other, and the length of the common chord is 12 cm. Find the distance between their centres.

Q.

Question 2
Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Q.

Question 1
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. In the given figure, AB is a chord of the circle with centre O and BOC is the diameter. If $OD\perp AB$ such that $OD=6\text{cm}$, then find AC.

1. $12\text{cm}$
2. $7.5\text{cm}$
3. $15\text{cm}$
4. $9\text{cm}$

Q. In the given figure, BOC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD. If AB = 10 cm, then CD = ?
(a) 5 cm
(b) 12.5 cm
(c) 15 cm
(d) 10 cm

Q. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
[3 Marks]

Q. Two parallel chords of lengths 30 cm and 16 cm are drawn on the opposite sides of the centre of a circle of radius 17 cm. Find the distance between the chords.

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

Q. Question 3
Write True or False and justify your answer :

The congruent circles with centres O and O' intersect at two points A and B. Then $\angle AOB=\angle A{O}^{\prime }B$.

Q.

Two chords AB and CD of lengths 5 cm 11cm 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.

Q.

Question 2
If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Q. In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is

(a) 34 cm

(b) 15 cm

(c) 23 cm

(d) 30 cm

Q. In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD, prove that AB = CD.

Q.

Question 3
If 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.