Theorem of Equal Chords Subtending Angles at the Center

Q.

If $AB,BC,$ and $CD$ are equal chords of a circle with $O$ as center, and $AD$ diameter then $\angle AOB$ equals to

1. None of these

2. ${60}^{\circ }$

3. ${90}^{\circ }$

4. ${120}^{\circ }$

Q.

The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.

Q. If chords AB and CD of congruent circles subtend equal angles at their centres, then which of the option is correct?

1. 
   $AB=CD$
2. 
   $AB>CD$
3. 
   $AB<CD$
4. 
   None of the above

Q.

In the adjoining figure, $DE$ is a chord parallel to diameter $AC$ of the circle with centre $O$. If $\angle CBD={60}^{\circ }$, calculate $\angle CDE.$

Q. Seg PM and seg PN are congruent chords of a circle with centre C. Show that the ray PC is the bisector of $\angle$NPM.

Q.

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

1. 115
2. 50
3. 100

4. 95

Q.

In the figure, AB is a diameter of a circle with centre O and CD. If $\angle$BAC = ${20}^{\circ }$, find the values of (i)$\angle$BOC (ii)$\angle$DOC (iii)$\angle$DAC (iv)$\angle$ADC

Q.

Question 6
Salil wants to put a picture in a frame. The picture is $7\frac{3}{5}$cm wide. To fit in the frame the picture cannot be more than $7\frac{3}{10}$cm wide. How much should the picture be trimmed?

Q. In figure ABCD is a cyclic quadrilateral; O is the centre of the circle. If $\angle BOD={160}^{\circ }$, find the measure of $\angle BPD$ , as shown in the given figure.

Q.

In the given figure, O is the centre of a circle in which chords AB and CD intersect at P such that PO bisects $\angle$BPD. Prove that AB = CD.

Q.

If the angles subtended by the chords of a circle at the centre are equal, then the chords are

1. Not equal to each other

2. Parallel to each other

3. Equal to each other

4. Perpendicular to each other

Q.

A sector of a circle with a radius $18\mathrm{cm}$ has a central angle $120°$. Find the area of the sector. Use $\mathrm{\pi }=3.14$

Q. In the given figure, ABC is isosceles triangle with $AB=AC,\angle BCD={20}^{\circ }.$ The measure of $\angle ACB$ is equal to

1. 
   ${100}^{\circ }$
2. 
   ${80}^{\circ }$
3. 
   ${60}^{\circ }$
4. 
   ${70}^{\circ }$

Q.

If two chords of a circle are equal, then their corresponding arcs will be equal.

1. False

2. True

Q.

Which statements are true regarding the area of circles and sectors? Check all that apply.

1. The area of a circle depends on the length of the radius.

2. The area of a sector depends on the ratio of the central angle to the entire circle.

3. The area of a sector depends on $\mathrm{\pi }$.

4. The area of the entire circle can be used to find the area of a sector.

5. The area of a sector can be used to find the area of a circle.

Q.

In figure, A, B, and C are three points on a circle with centre O such that $\angle$BOC = ${30}^0$ and $\angle$AOB = ${60}^0$. If D is a point on the circle other than the arc ABC, find $\angle$ADC (in degrees).

__

Q.

The minute hand of a circular clock is $15cm$ long. How far does the tip of the minute hand move in $1$ hour. (Take $\mathrm{\pi }=3.14$)

Q.

Show that the tangents at the end points of a diameter of a circle are parallel.

Q. In the given figure, O is the centre of the circle of radius 10 cm. AB and AC are two chords such that

$AB=AC=4\sqrt{5}cm$.

The length of chord BC is

1. 
   12.4 cm
2. 
   24 cm
3. 
   8 cm
4. 
   16 cm

Q. If $AB=CD$ and $\angle AOB={50}^{\circ }$, then $\angle ODC$ = _____ .

1. ${65}^{\circ }$
2. ${50}^{\circ }$
3. ${75}^{\circ }$
4. ${85}^{\circ }$

Q.

Equal chords subtend equal angles at the centre.

1. True
2. False

Q.

Question 2
A chord AB of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the major arc and also at a point on the minor arc.

Q.

In the following figure, what can be concluded from the below given conditions.

Given: AB = CD and AC = DE.

1. All of the above.

2. AC = BD, $\angle$AOC = $\angle$BOD

3. $\angle$AOC = $\angle$DOE, AC = DE

4. AD = CE, $\angle$AOD = $\angle$COE.

Q. In the figure, AOB is a diameter of the circle and C, D, E are any three points on the semi-circle. Find the value of $\angle ACD+\angle BED$.

Q. AB and PQ are two equal chords of a circle. If the length of minor arc and major arc formed by AB is 24cm and 64cm, respectively, then find the length of minor and major arc formed by PQ.

Q.

Two circles of radius 3 cm and 5cm have a common centre, O. AB is a chord to both the circles and length of CD is $2\sqrt{5}$ cm. Find the distance of the chord from the centre and the length AC.

Q. In the figure, $\angle ADC={130}^{\circ }$ and chord BC = chord BE. Find $\angle CBE$

Q.

Two circles of radius 3 cm and 5cm have a common centre, O. AB is a chord to both the circles and length of CD is $2\sqrt{5}$ cm. Find the distance of the chord from the centre and the length AC.

Q. A chord of length 6 cm is drawn on a circle of radius 5 cm. Calculate its distance from the centre of the circle.

Q. Find the value of x in the given figure if$\angle AOB=70°$ and AB = CD.

1. 70°
2. 80°
3. 50°
4. 60°