# Circle

## Key Points

**Circle:** The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

**Theorem 1:** Equal chords of a circle subtend equal angles at the centre.

**Theorem 2:** If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

**Theorem 3:** The perpendicular from the centre of a circle to a chord bisects the chord.

**Theorem 4:** The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

**Theorem 5:** There is one and only one circle passing through three given non-collinear points.

**Theorem 6:** Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres)

**Theorem 7:** Chords equidistant from the centre of a circle are equal in length.

**Theorem 8:** The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

**Theorem 9:** Angles in the same segment of a circle are equal.

**Theorem 10:** If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic)

**Theorem 11:** The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.

**Theorem 12:** If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

## Exercise 10.1

Question 1: Fill in the blanks:

- The centre of a circle lies in …………..of the circle. (exterior/ interior)
- A point, whose distance from the centre of a circle is greater than its radius lies in …………of the circle. (exterior/ interior)
- The longest chord of a circle is a ………..of the circle.
- An arc is a ………….when its ends are the ends of a diameter.
- Segment of a circle is the region between an arc and ……………of the circle.
- A circle divides the plane, on which it lies, in …………….parts.

**Answer:** (i) interior, (ii) exterior, (iii) diameter, (iv) semicircle, (v) chord, (vi) three

Question 2: Write True or False

(i) Line segment joining the centre to any point on the circle is a radius of the circle.

**Answer:** True; because the distance between centre and any point on circumference is called radius.

(ii) A circle has only finite number of equal chords.

**Answer:** False, because a circle can haev infinite number of equal chords.

(iii) If a circle is divided into three equal arcs, each is a major arc.

**Answer:** False, because a major arc is always bigger than a minor arc, and they cannot be equal.

(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.

**Answer:** True because diameter is the longest chord of a circle and is always twice the radius.

(v) Sector is the region between the chord and its corresponding arc.

**Answer:** False, the region between the chord and its corresponding arc is called segment.

(vi) A circle is a plane figure.

**Answer:** True