Equal Chords Are at Equal Distances from the Center

Q. The chord of a circle is equal to its radius. The angle subtended by this chord at the minor arc of the circle is

(a) 60°

(b) 75°

(c) 120°

(d) 150°

Q. Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circle is

(a) $\sqrt{r}$

(b) $\sqrt{2}rAB$

(c) $\sqrt{3}r$

(d) $\frac{\sqrt{3}}{2}$

Q. Question 5
If a line segment joining midpoints of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

Q.

Two chords $AB$ and $AC$ of a circle subtends angles equal to $90°$and $150°$, respectively at the centre. Find $\angle BAC$, if $AB$ and $AC$ lie on the opposite sides of the centre.

Q.

An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.

Q. Which of the following is/are correct?

1. The perpendicular drawn from the centre of the circle to a chord bisects the chord.
2. Congruent arcs of a circle subtend equal angles at the centre.
3. Congruent arcs of a circle subtend right angles at the centre.
4. Chords equidistant from the centre of a circle are equal in length.

Q.

If two equal chords are drawn on a circle then they are equidistant from the centre of circle.

1. False

2. True

Q. In the given circle with centre ‘O’, the mid – points of two equal chords $AB$ & $CD$ are $K$ & $L$ respectively.



If $\angle OLK={25}^{\circ }$, then $\angle LKB=$?

1. ${125}^{\circ }$
2. ${115}^{\circ }$
3. ${90}^{\circ }$
4. ${105}^{\circ }$

Q. In the given figure, O is the centre of a circle. AB and CD are its two chords. If OM $\perp$ AB and ON $\perp$ CD and OM = ON, then

1. $AB\ne CD$
2. $AB<CD$
3. $AB>CD$
4. $AB=CD$

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. Plz tell me the difference between the theorem 10.1 equal chords of a circle subtend equal at the centre and theorem 10.6 equal chords of a circle (or congruent circle) subtend equal angles at the centre of the circle .plz state how the two theorems r different .and proof it

Q.

AB and CD are equal chords of a circle whose centre is O. When produced, these chords meet at E. Prove that EB = ED. [3 MARKS]

Q.

Two equal chords, $AB$ and $CD$ are at a distance of 10 cm from each other. Find the distance of chord $AB$ from the centre.

1. $6cm$

2. $5cm$

3. $20cm$

4. $12cm$

Q.

AB & CD are two equal chords of a circle with Centre O which intersect each other at the right angle 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. kite.

3. rhombus.

4. rectangle.

Q.

AB & CD are two equal chords of a circle with Centre O, which intersect each other at a right angle 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. rectangle.

3. rhombus.

4. kite.

Q.

Chords of a circle, equidistant from the centre are

1. Parallel

2. Equal

3. Perpendicular

4. Not equal

Q. In a circle, $AB$ and $CD$ are two parallel chords with centre $O$ and radius $10cm$ such that $AB=16cm$ and $CD=12cm$ determine the distance between the two chords?

Q.

Question 5
If a line segment joining midpoints of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

Q. Two parallel chords are drawn on the same side of the centre of a circle of radius 20. It is found that they subtend ${60}^0$ and ${120}^0$ angles at the centre of the circle. Then the perpendicular distance between the chords is:

1. $5(\sqrt{3}-1)$
2. $10(\sqrt{2}-1)$
3. $10(\sqrt{3}-1)$
4. $5(\sqrt{3}+1)$

Q. If chords 𝐴𝐵 and 𝐶𝐷 subtend angles at the centre 𝑂 of the circle such that ∠𝐴𝑂𝐵 = ∠𝐶𝑂𝐷, then:

1. $AB=CD$
2. $AB>CD$
3. $AB<CD$

Q. In the given figure, OP and OQ are perpendicular bisectors of chords CD and AB respectively. If PD = 4 cm and OP = OQ = 3 cm, then what will be the length of chord AB?

1. 4 cm
2. 5 cm
3. 8 cm
4. 6 cm

Q. If $\angle OAB=40°$ and $AB=CD$, then find the measure of $\angle COD$.

Q. In the given figure, OP and OQ are perpendicular bisectors of chords CD and AB respectively. If PD = 4 cm and OP = OQ = 3 cm, then calculate the length of chord AB.

Q. In a circle of radius 13 𝑐𝑚, two chords $MN$ and $JK$ are at a distance of 5 𝑐𝑚 from the centre. Find the measure of $(MN+2\times JK)$ .

Q. In the given circle with centre ${}^{\prime }{O}^{\prime }$, the mid points of two equal chords $AB$ & $CD$ are $K$ & $L$, respectively. If $\angle$ $OLK=$ ${25}^{\circ }$, Then $\angle$ $LKB$ is equal to:

1. ${125}^{\circ }$
2. ${90}^{\circ }$
3. ${105}^{\circ }$
4. ${115}^{\circ }$

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 ⊥ AB and
OQ ⊥ AC, prove that PB = QC.

[3 marks]
[RS Agrawal]
[Equal Chords of a Circle are Equidistant from the Centre]

Q. In fig 4.44 chord AB = chord CD, seg AB $\perp$ seg CD. Points M and N are midpoints of chord AB and chord CD respectively prove that $\square$MONE is a square.

Q. In the given figure, AB and CD are two equal chords of the circle with centre O. OP and OQ are perpendiculars of the chord AB and CD respectively. If $\angle POQ={80}^{\circ },$ then what is the value of $\angle APQ$?

1. ${60}^{\circ }$
2. ${50}^{\circ }$
3. ${30}^{\circ }$
4. ${40}^{\circ }$

Q. In the adjoining figure, chords AC and BD of a circle with centre O, intersect at right angles at E. If ∠OAB = 25°, calculate ∠EBC.

Q. Equal chords are equidistant from the centre of circle.

1. False
2. True