The Circle

Given: O is a centre of a circle.

Firstly, we need to know what is radius, diameter, chord?

Radius: Any line segment connecting the centre of a circle to any point of a circle.

Chord: A line segment joining any two points of a circle.

Diameter: The chord of a circle which passes through the centre of the circle.

This means, diameter is the longest chord of a circle.

Also, diameter of a circle = 2 r, where r = radius

From the definition defined above and by the given figure in question we can answer the given question

(1) Seg OD is a radius

(2) Seg AB is a diameter

(3) SEg PQ is a chord

(4) The length of seg AB is twice that of seg OD.

Explanation: seg OD is a radius and seg AB is a diameter and we have seen above that diameter is twice of radius.

Firstly, we need to know what is centre, radius, diameter, chord?

Centre: The centre of a circle is a point.

Radius: Any line segment connecting the centre of a circle to any point of a circle.

Chord: A line segment joining any two points of a circle.

Diameter: The chord of a circle which passes through the centre of the circle.

This means, diameter is the longest chord of a circle.

Also, diameter of a circle = 2 r , where r = radius

From the definition defined above and by the given figure in question we can answer the given question

Centre of the given figure is C

Radius of the given figure is seg CL, seg CM, seg CD

Diameter of the given figure is seg LM

Chord of the given figure is seg TR, seg LM (since diameter is the longest chord)

1. False,

Reason: a line segment joining any two points of a circle is a chord, thus seg TS is a chord

2. True,

Reason: KM is a diameter, and diameter is also known as the longest chord of a circle.

3. True,

Reason: C is a centre and any line segment connecting the centre of a circle to any point of a circle is a radius, Thus, CK is a radius.

4. False,

Reason: seg KM is a diameter since it is a chord passing through the centre C of a circle.

Given:

Chord RT is at a distance of from centre A.

Draw a figure using given condition

Let us draw a perpendicular bisector AC to MN and AB to RT and

To Show: AB = AC

Join AT and AN, we get,

Now in ∆ABT and ∆ACN

RT = MN (since, chord MN ≅ chord RT )

BT = CN

∠ABT = ∠ACN = 90^°

seg AT = seg AN (since, AT and AN are radius of a circle)

Therefore, ∆ABT ≅ ∆ACN (By RHS Congruent Rule)

Therefore, seg AC = seg AB = 6cm (By CPCT)

Alternate Method: Given: chord MN ≅ chord RT

Chord RT is at a distance of from centre A.

To Find length of seg AC

We know that, Congruent chord are equidistant from the centre.

Therefore, seg AC = seg AB = 6cm

Given: seg 0M ⊥ seg AB

That is ∠OMB = ∠OMA = 90°

seg AM = 1.5cm

Join OA and OB, we get,

Now in ∆OMA and ∆OMB

∠OMB = ∠OMA = 90° (since, seg 0M⊥seg AB )

seg OA = seg OB (since, OA and OB are radius of a circle)

seg OM = seg OM (since, common side)

Therefore, ∆OMA ≅ ∆OMB (By, RHS Congruent Rule)

Therefore, seg AM = seg BM (By CPCT)

Thus, seg AM = seg BM = 1.5cm

seg AB = seg AM + seg BM

seg AB = 1.5 + 1.5 = 3cm

Given: chord AB ≅ chord CD

AB = CD

m∠APB = 40°

We have a circle with centre P, Let us now draw a figure with the given information.

Join, PC and PD, we get,

Now, In ∆APB and ∆PCD

AB = CD (since, chordAB≅ chord CD )

BP = PD (since, BP and PD are radius of a circle)

AP = PC (since, AP and PC are radius of a circle)

Thus, ∆APB≅∆PCD (By, SSS Congruent Rule)

Thus, m∠APB = m∠CPD = 40° (By CPCT)

Given: radius of a circle = 5cm

Distance of chord from centre = 4cm

By, Given information we will draw a circle

Given, radius PB = 5cm

PC = 4cm where, P is perpendicular bisector of segAB

Thus, ∠PCB = 90°

By Pythagoras formula

PB² = BC² + PC²

5² = BC² + 4²

25 = BC² + 16

BC² = 25 - 16 = 9cm

BC = √9 = 3cm

Since, P is perpendicular bisector of seg AB

Therefore, BC = CA = 3cm

Therefore, BA = BC + CA

BA = 3 + 3 = 6cm

Thus, length of a chord BA = 6cm

Given: radius of a circle = 13cm

Length of chord = 10cm

By, Given information we will draw a circle.

Given: radius AB = 13cm

chord BC = 10cm

Let, us draw a perpendicular bisector AD of BC

Thus,

Also, since AD is perpendicular bisector of BC

Therefore,

By Pythagoras formula

Thus, distance of the chord from the centre of the circle is 12cm

Given: distance of the chord from the centre =

Length of chord

By, Given information we will draw a circle.

Here, distance of the chord from the centre

Length of chord

Where, AD is perpendicular bisector of chord BC

Thus,

Join, AB, we get

Since,

Therefore, we will use By Pythagoras formula

Thus, radius of a circle