Circles

In this chapter, you will be studied the following points:

1. A circle is the collection of all points in a plane, which are equidistant from a fixed point in the plane.

2. Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.

3. If the angles subtended by two chords of a circle (or of congruent circles) at the centre (corresponding centres) are equal, the chords are equal.

4. The perpendicular from the centre of a circle to a chord bisects the chord.

5. The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

6. There is one and only one circle passing through three non-collinear points.

7. Equal chords of a circle (or of congruent circles) are equidistant from the centre (or corresponding centres).

8. Chords equidistant from the centre (or corresponding centres) of a circle (or of congruent circles) are equal.

9. If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two chords of a circle are equal, then their corresponding arcs (minor, major) are congruent.

10. Congruent arcs of a circle subtend equal angles at the centre.

11. 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.

12. Angles in the same segment of a circle are equal.

13. Angle in a semicircle is a right angle.

14. 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.

15. The sum of either pair of opposite angles of a cyclic quadrilateral is 180⁰.

16. If sum of a pair of opposite angles of a quadrilateral is 1800, the quadrilateral is cyclic.

For details of the above learning points join me on my live class in "YouTube" on every Sunday.

Linear Equation in two variables

Linear Equation in two variables

A concise details accordingly class 9.

1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables.

2. A linear equation in two variables has infinitely many solutions.

3. The graph of every linear equation in two variables is a straight line.

4. x = 0 is the equation of the y-axis and y = 0 is the equation of the x-axis.

5. The graph of x = a is a straight line parallel to the y-axis.

6. The graph of y = a is a straight line parallel to the x-axis.

7. An equation of the type y = mx represents a line passing through the origin.

8. Every point on the graph of a linear equation in two variables is a solution of the linear equation. Moreover, every solution of the linear equation is a point on the graph of the linear equation.