Circles

We know that,

Diameter = 2 × Radius

Given,

Radius = 5 cm

⇒ Diameter = 2 × 5 cm

⇒ Diameter = 10 cm

Hence, the length of the diameter = 10 cm.

We know that,

Diameter = 2 × Radius

Given,

Radius = 3.6 cm

⇒ Diameter = 2 × 3.6 cm

⇒ Diameter = 7.2 cm

Hence, the length of the diameter = 7.2 cm.

We know that,

Diameter = 2 × Radius

Given,

Radius = 2.8 cm

⇒ Diameter = 2 × 2.8 cm

⇒ Diameter = 5.6 cm

Hence, the length of the diameter = 5.6 cm.

(i) The perimeter of a circle is called its circumference.

(ii) A diameter is the largest chord of the circle.

(iii) Half of a circle is called a semi-circle.

(iv) The mid-point of a diameter of a circle is the centre of the circle.

(v) All the radii of a circle are equal.

(vi) A line segment having its end points on a circle is called a chord of the circle.

(vii) A chord that passes through the centre of a circle is called a diameter of the circle.

(viii) The interior of a circle together with the circle is called the circular region.

(ix) A line which intersects the circle at two distinct points is called a secant.

(x) A quadrant is one-fourth of a circle.

We know that,

A circle is a simple closed curve consisting of all points in a plane which are at a fixed distance from a fixed point inside it.

A circle is made up of a curved line and not of line segments, so it is not a polygon.

Hence, option 3 is the correct option.

We know that,

The whole arc of a circle is called its circumference. So, the length of the boundary of a circle is called its circumference.

Hence, option 1 is the correct option.

We know that,

A point P lies in the interior of a circle with centre O and radius r, if OP < r.

The distance of the centre O from itself is 0, which is less than the radius r.

So, the centre of a circle lies in its interior.

Hence, option 1 is the correct option.

We know that,

A line segment joining any point on the circle to its centre is called a radius of the circle.

Hence, option 2 is the correct option.

We know that,

Circles having the same centre but different radii are called concentric circles.

Hence, option 2 is the correct option.

We know that,

A line segment whose end points lie on a circle is called a chord of the circle.

Hence, option 2 is the correct option.

We know that,

A line which intersects the circle at two distinct points is called a secant.

Hence, option 2 is the correct option.

We know that,

Any part of a circle is called an arc.

Hence, option 2 is the correct option.

We know that,

A chord of a circle divides its circular region into two parts. Each part is called a segment of the circle.

Hence, option 2 is the correct option.

Given,

Radius = 5 cm.

We know that,

Diameter = 2 × Radius

⇒ Diameter = 2 × 5 cm

⇒ Diameter = 10 cm

Hence, option 4 is the correct option.

Given,

Diameter = 8.5 cm.

We know that,

Diameter = 2 × Radius

⇒ Radius = $\frac{\text{Diameter}}{2}$

⇒ Radius = $\frac{8.5}{2}$ cm

⇒ Radius = 4.25 cm

Hence, option 2 is the correct option.

(i) The length of the boundary of a circle is called its circumference.

(ii) The centre of a circle lies in its interior.

(iii) Radius of a circle is half of its diameter.

(iv) An arc of a circle is a part of the circle.

(v) All the radii of circle are equal.

(vi) If O is the centre of a circle and two radii OA and OB are at right angles then the sector OARBO is called the quadrant of the circle.

(i) One-half of the whole arc of a circle is called a semi-circle.

Hence, the given statement is true (T).

(ii) A major arc is only a part of the whole arc of the circle, and the whole arc of a circle is its circumference.

⇒ (Length of major arc) < (Circumference of the circle)

Hence, the given statement is false (F).

(iii) An arc less than one-half of the whole arc of a circle is called a minor arc, and one-half of the whole arc of a circle is a semi-circle.

⇒ (Length of minor arc) < (Length of the semi-circle)

Hence, the given statement is true (T).

(iv) The whole arc of a circle is called its circumference.

Hence, the given statement is true (T).

(v) A chord of a circle divides its circular region into two parts and each part is called a segment of the circle.

Hence, the given statement is true (T).

1. From figure,

OB = 6.5 cm, where O is the centre of the circle and B is a point on the circle.

So, OB is a radius of the circle.

We know that,

Diameter = 2 × Radius

⇒ Diameter = 2 × 6.5 cm

⇒ Diameter = 13 cm

Hence, option (d) is the correct option.

2. We know that,

An arc greater than one-half of the whole arc of a circle is called a major arc of the circle.

From figure, the arc starting from A and going through E, F, B and J to reach C is greater than one-half of the whole arc of the circle.

So, $\widehat{AFC}$is a major arc of the circle.

Hence, option (c) is the correct option.

3. We know that,

A line segment whose end points lie on a circle is called a chord.

From figure, the end points of AB, CD and EF lie on the circle, so they are chords of the circle.

In OA, the end point O is the centre of the circle, which lies in the interior of the circle and not on the circle. So, OA is a radius and not a chord of the circle.

Hence, option (b) is the correct option.

4. We know that,

The part of the circular region bounded by an arc and the two radii at the ends of the arc is called a sector, and the sector containing the minor arc is called the minor sector.

From figure, the radii OC and OB together with the minor arc CJB bound the smaller part of the circular region.

So, OCJBO is a minor sector of the circle.

Hence, option (b) is the correct option.

5. We know that,

A line which touches a circle at one point only, is called a tangent to the circle at that point.

From figure, OD is a radius and CD is a chord of the circle, whereas the line GH touches the circle at the point D only.

So, GH is a tangent to the circle at D.

Hence, option (c) is the correct option.

By joining any point on the circle with its centre, we get a radius. Since a circle has an infinite number of points, we can draw an infinite number of radii of a circle.

So, Assertion (A) is true.

A chord of a circle passing through its centre is called a diameter of the circle.

So, Reason (R) is true.

Reason (R) gives the definition of a diameter and does not tell why an infinite number of radii can be drawn. So, Reason (R) is not the correct explanation of Assertion (A).

∴ Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.

A chord of a circle divides its circular region into two parts and each part is called a segment of the circle.

So, Assertion (A) is true.

One-half of the whole arc of a circle is called a semi-circle.

So, Reason (R) is true.

Reason (R) gives the definition of a semi-circle and does not explain the meaning of a segment. So, Reason (R) is not the correct explanation of Assertion (A).

∴ Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

Hence, option 2 is the correct option.