NCERT Solutions for Class 9 Maths Chapter 9 Circles

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

1. Diameter: It is the longest chord of the circle.
2. Circumference: The length of complete circle is called its circumference.
3. Arc: A piece of circle between two point is called arc.
4. Segment: The region between a chord and either of its arcs is called a segment of circular region.

Properties related to CIRCLE
1. Equal chords of a circle subtend equal angles at the centre.
2. If the angles subtended by two chords of a circle at the centre are equal, the chords are also equal.
3. The perpendicular from the centre of a circle to a chord bisects the chord.
4. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
5. There is one and only one circle passing through three non-collinear points.

AC is diameter of circle. Hence, ∠ADC = 90° and ∠ABC = 90° … (1) [∵ Angle in a semicircle is a right angle.] Similarly, BD is diameter of circle. Hence, ∠BAD = 90° and ∠BCD = 90° … (2) [∵ Angle in a semicircle is a right angle.] From the equation (1) and (2), ∠ADC =∠ABC = ∠BAD = ∠BCD = 90° Hence, ABCD is a rectangle.

The centre of a circle lies in interior of the circle.

False. Because, there are infinite number of equal chords in a circle.

Sector is the region between the chord and its corresponding arc. False. Because, between chord and arc a segment is formed. Sector is the region which is formed between radii and arc.

Given: Triangle ABC and ADC are two right triangle on common base AC.
To prove: ∠CAD = ∠CBD.
Proof: Triangle ABC and ADC are on common base BC and ∠BAC = ∠BDC.
Hence, points A, B, C and D lie on the same circle.
[∵ 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.]
Therefore, ∠CAD = ∠CBD [∵ Angles in the same segments are equal]