10th Class Mathematics Circles

Circles

Category : 10th Class

Circles

• Secant: A line which intersects a circle at two distinct points is called a secant of a circle.

• Tangent: A line touching a circle at exactly one point only is called a tangent to the circle at that point.

• Point of contact: The point P at which the tangent touches the circle is called the point of contact.
• Number of tangents to a circle:
 Position of the point w.r.t. the circle Number of tangents Inside 0 On 1 Outside 2

• Length of a tangent: The length of the line segment of the tangent between a given point and the given point of contact with the circle is called the length of the tangent from the point to the circle.

The tangent at any point of a circle is perpendicular to the radius through the point of contact. In other words, the angle between a tangent and the radius through the point of contact is${{90}^{o}}$.

• The lengths of tangents drawn from an external point to a circle are equal.

If AP and AQ are two tangents from an external point A to the circle, then AP = AQ.

• Two tangents drawn from an external point subtend equal angles at the centre and are equally inclined to the line segment joining the centre to that point.

• The tangents drawn at the ends of a diameter of a circle are parallel.

• The line segments joining the point of contact of two parallel tangents to a circle is a diameter of the circle.
• The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by  the line segments joining the points of contact to the centre.

• There is one and only one tangent at any point on the circumference of a circle.

• A parallelogram circumscribing a circle is a rhombus.
• The opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

$\angle AOB+\angle COD={{180}^{o}}$

$\angle BOC+\angle AOD={{180}^{o}}$

• In two concentric circles, the chord of the larger circle which touches the smaller circle is bisected at the point of contact.

$\text{AP}=\text{PB}$

• If PAB is a secant to a circle intersecting it at A and B and PT is a tangent, then$\text{PA}\times \text{PB}=\text{P}{{\text{T}}^{\text{2}}}$.

Other Topics

Notes - Circles

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