A brief introduction to circles for class 10 is provided here. Get the complete description provided here to learn about the concept of the circle. Also, learn how to draw a tangent to the circle with various theorems and examples.
Introduction to Circles
Circle and line in a plane
For a circle and a line on a plane, there can be three possibilities.
i) they can be non-intersecting
ii) they can have a single common point – in this case, the line touches the circle.
ii) they can have two common points – in this case, the line cuts the circle.
A tangent to a circle is a line which touches the circle at exactly one point. For every point on the circle, there is a unique tangent passing through it.
A secant to a circle is a line which has two points in common with the circle. It cuts the circle at two points, forming a chord of the circle.
Tangent as a special case of Secant
The tangent to a circle can be seen as a special case of the secant, when the two end points of its corresponding chord coincide.
Two parallel tangents at most for a given secant
For every given secant of a circle, there are exactly two tangents which are parallel to it and touches the circle at two diametrically opposite points.
Tangent perpendicular to the radius at the point of contact
Theorem: The theorem states that “the tangent to the circle at any point is the perpendicular to the radius of the circle that passes through the point of contact”.
Here, O is the centre and OP⊥XY.
The number of tangents drawn from a given point
i) If the point is in an interior region of the circle, any line through that point will be a secant. So, no tangent can be drawn to a circle which passes through a point that lies inside it.
ii) When a point of tangency lies on the circle, there is exactly one tangent to a circle that passes through it.
iii) When the point lies outside of the circle, there are accurately two tangents to a circle through it
Length of a tangent
The length of the tangent from the point (Say P) to the circle is defined as the segment of the tangent from the external point P to the point of tangency I with the circle. In this case, PI is the tangent length.
Lengths of tangents drawn from an external point
Theorem: Two tangents are of equal length when the tangent is drawn from an external point to a circle.