Theorem 2: Perpendicular from the Center to a Chord Bisects the Chord

Q.

The circle inscribed in an equilateral triangle of sides 24 cm just touches the sides of the triangle. Find the area of the rest part of the triangle (let $\sqrt{3}$=1.732)

1. 93 sq.cm

2. 98.54 sq.cm

3. 92.70 sq.cm

4. 96.67 sq.cm

Q.

Two circles of radii $4cm$ and $3cm$ intersect at two points and the distance between their centres is $5cm$. Find the length of the common chord.

1. $4.8cm$

2. $2.8cm$

3. $7cm$

4. $5.5cm$

Q.

In the given figure, O is the centre of the circle. Find $x$.

1. ${50}^{\circ }$

2. ${40}^{\circ }$

3. ${20}^{\circ }$

4. ${30}^{\circ }$

Q.

In figure $A,B,C$ are three points on a circle such that the angles subtended by the chord $AB$ and $AC$ at the centre $O$ are ${80}^{\circ }$and ${120}^{\circ }$ respectively. Determine $\angle BAC$ and the degree measure of arc $BPC$.

Q.

Equal chords AB and CD of a circle with centre O, cut at right angles at E. If M and N are the mid - points of AB and CD respectively, then OMEN is a __.

Q. If the line from the centre of a circle to a chord (other than the diameter) bisects the chord, then the angle between the chord and the line is:

1. ${30}^{\circ }$
2. ${60}^{\circ }$
3. ${90}^{\circ }$
4. ${45}^{\circ }$

Q.

Two circles of radii $5cm$ and $6cm$ with common centre are drawn. There is a line $AB$ such that it is chord to both the circles. $CD=8cm.$ Find the distance of the chord from centre and the length of $AC$ respectively.

1. $3cm,1.19cm$

2. $2.55cm,2.35cm$

3. $3cm,2cm$

4. $2.16cm,2.35cm$

Q. In the given figure, O is the centre of the circle and $\angle AOC={120}^{\circ }$. Find $\angle BDC$.

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${15}^{\circ }$

Q.

In the figure, $O$ is the centre of the circle. If $\angle BAC={52}^{\circ }$, then $\angle OCB$ is equal to

1. 52

2. 104

3. 38

4. 76

Q.

If two arcs of equal length subtend angles x and y at the centre, then x = __

1. y

2. 2y

3. 5y

4. 3y

Q.

AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on opposite sides of the centre and the distance between them is 17 cm, find the radius of the circle.

1. 13 cm

2. 10 cm

3. 11 cm

4. 12 cm

Q.

In two concentric circles with centre O, if AD is the chord of the larger circle, then AB = CD.

1. True

2. False

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

1. ${120}^o$
2. ${240}^o$
3. ${60}^o$
4. ${30}^o$

Q. Question 6
In the figure, if $\angle OAB={40}^{\circ }$, the $\angle ACB$ is equal to


(A) ${50}^{\circ }$
(B) ${40}^{\circ }$
(C) ${60}^{\circ }$
(D) ${70}^{\circ }$

Q. In the given figure, if O is the centre of the circle, then find the measure of $\angle OCD+\angle ODC.$

1. ${100}^{\circ }$
2. ${110}^{\circ }$
3. ${120}^{\circ }$
4. ${130}^{\circ }$

Q.

If two equal chords AB and CD of a circle when produced, intersect at point P, then PB =

1. PD

2. CY

3. None of the above

4. DY

Q.

The length of arc AB is twice the length of arc BC of a circle with centre O. If $\angle$AOB is ${100}^o$ then $\angle$BOC is

1. ${50}^o$

2. ${60}^o$

3. ${70}^o$

4. ${55}^o$

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 other chord.

Q.

A chord AB of a circle whose centre is O, is bisected at E by a diameter CD. OC= OD = 15 cm and OE = 9 cm. Find the length of AD.

1. 60°

2. 70°

3. 80°

4. 90°

Q. Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Q.

In the given figure $\angle$ AOC=${130}^{\circ }$, The value of $\angle$ ABC (in degrees) is _______.

1. ${115}^{\circ }$

2. $50\circ$

3. ${100}^{\circ }$

4. ${95}^{\circ }$

Q.

It is possible for two chords to not be equal even if they are at a same distance from the centre when

1. the chords are parallel to each other

2. the chords are perpendicular to each other

3. the chords are drawn to concentric circles

4. None of these

Q. In the given figure, O is the centre of the circle and ∠AOC = 120°. Find ∠BDC.

1. 45°
2. 30°
3. 15°
4. 60°

Q.

The perpendicular to a chord from the centre of the circle divides the chord in ratio of

1. $1:1$

2. $1:3$

3. $1:2$

4. $1:4$

Q. In the given figure, O is the centre of the circle. If ∠AOB = 100° and ∠AOC = 90°, then find ∠BAC.

1. 80°
2. 90°
3. 75°
4. 85°

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

Q.

Check whether the following statement is true or false:

If AB and AC are two chords of a circle with centre O, then we can conclude that ∠AOB=∠AOC.

Q. If $t_1$ and $t_2$ are two points on the parabola $y^2=4ax$. If the focal chord joining them coincides with the normal chord, then

1. $t_1+t_2=0$
2. $t_1(t_1+t_2)+2=0$
3. $t_1{t}_2=-1$
4. None of these

Q.

S1: $AM=\frac{AP}{2},BN=\frac{BP}{2}$

S2: $O{O}^{\prime }=AM+BN$

Choose the correct option.

1. Only S1 is true.

2. S1 is true & S2 is false.

3. Both statements are true.

4. Only S2 is true.