# Theorem 1: Equal Chords Subtend Equal Angles at the Center

## Trending Questions

**Q.**In the given pentagon ABCDE, AB = BC = CD = DE = AE. What is the value of x?

- 36∘
- 54∘
- 108∘
- 72∘

**Q.**

A circle touches the $y$-axis at the point $(0,4)$and cuts the $x$-axis in a chord of length $6$ units. The radius of the circle is

$3$

$4$

$5$

$6$

**Q.**

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

**Q.**AB = 36 cm and CD = 48 cm are two parallel chords of a circle with centre O. The distance between them is 42 cm. The radius of the circle is:

- 27cm
- 25 cm
- 30 cm
- 33 cm

**Q.**

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

5 cm

6 cm

7 cm

8 cm

**Q.**

If (x+y)^{-1}(x^{-1}+y^{-1})= x^{p} y^{q}, then p+q is equal to

2

-1

-2

1

**Q.**

**Question 1**

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

**Q.**A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

**Q.**

In the given figure, △ABC is an isosceles triangle with AB=AC and O is the centre of the circle. If ∠AOB=α and ∠AOC=β, then which of the following is true?

α=2β

α=β2

α=β

α=3β

**Q.**

AB and CD are the chords of a circle whose center is O. They intersect each other at P. If PO is the bisector of ∠APD, prove that AB = CD. [2 MARKS]

**Q.**

P and Q are points on intersection of two circles with centre O and O'. If straight line APB and CQD are parallel to OO' then AB =

**Q.**Angle subtended by one of the equal chords at centre is 60 degrees, angle made by other chord is:

- 30 degrees
- 60 degrees
- 12 degrees
- 45 degrees

**Q.**

If a line intersects two concentric circles (circles with the same centre) with centre $\mathrm{O}$ at $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$, prove that $\mathrm{AB}=\mathrm{CD}$ (see Fig) .

**Q.**

**Question 5**

In the given figure, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC=130∘ and ∠ECD=20∘. Find ∠BAC.

**Q.**Consider the below figure. Here, z = ___.

- 45∘
- 60∘
- 30∘
- 15∘

**Q.**Question 2

In figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB, then CD is equal to

(A) 2 cm

(B) 3 cm

(C) 4 cm

(D) 5 cm

**Q.**

Two chords AB, CD of length 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm. find the radius of the circle.Assume that the chords are on the same side of the center.

5.02cm

7.02cm

8.02cm

6.02cm

**Q.**

Two chords AB and AC of a circle subtends angles equal to 90∘ and 150∘ respectively at the centre. Find ∠BAC.

90∘

60∘

80∘

120∘

**Q.**

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

90∘

60∘

120∘

80∘

**Q.**Question 9

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

**Q.**

In the given figure, lengths of the chords AB and CD are 12 cm and 18 cm respectively and distance between them is 15 cm. Find the radius of the circle.

**Q.**

In the figure below, it is given that chords AB = PQ. The ∠AOB=550, and ∠POQ=x0, find x.

25

35

45

55

**Q.**

In the given figure, it is given that chords AB=PQ, ∠AOB=550, and ∠POQ=x0, find x.

55

- 35
- 45
- 25

**Q.**

In the given figure, chords AB=CD, ∠COD=60∘ and ∠AOD=170∘.

Find the value of ∠BOC.

60∘

70∘

75∘

80∘

**Q.**

Two arcs APB and CQD of a circle are in the ratio 5 :7. The angle subtended by arc APB at the centre is 150o.

1

2

**Q.**35. Two circles A and B of radius 3 cm and 4cm intersect at the point C and D such thatcAC and BC are tangent to the circle. Find the lenght of common chord CD

**Q.**

Given one line and one circle that you can arrange in any way you like, the minimum number of points where they intersect is

0

1

2

Infinity

**Q.**

A chord 24cm long is drawn in a circle of radius 13cm. Find its distance from the centre.

5 cm

50 cm

4 cm

14 cm

**Q.**Which of the following is/are true?

In a circle, if two chords are equal in measure, then their corresponding minor arcs are equal in measure.

- If two arcs of a circle are equal, the subtend equal angles at the centre.
- Circles having different radii are similar.
- Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

**Q.**O is the centre of the circle. ∠OAB=20∘, ∠OCB=55∘. Find ∠BOC and ∠AOC.