Circles Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Introduction to Circles:

   • A circle is a set of points in a plane that are equidistant from a fixed point called the center.
   • The radius is the distance from the center to any point on the circle.
   • The diameter is twice the radius and the longest chord of the circle.
   • A chord is a line segment joining any two points on the circle.
   • A secant is a line that intersects the circle at two points.
   • A tangent is a line that touches the circle at exactly one point.

2. Tangent to a Circle:

   • A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact.
   • The radius drawn to the point of contact is perpendicular to the tangent.
   • A circle can have infinitely many tangents.
   • Length of Tangent: The length of the tangent drawn from an external point to a circle is equal. If $PA$ and $PB$ are two tangents from a point $P$ to a circle with center $O$, then $PA = PB$.

3. Properties of Tangents:

   • Property 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
   • Property 2: The lengths of tangents drawn from an external point to a circle are equal.
   • Property 3: If two tangents are drawn from an external point to a circle, then the angle between the two tangents is supplementary to the angle subtended by the line segment joining the points of contact at the center of the circle.

4. Theorems Related to Circles:

   • Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
   • Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
   • Theorem 3: The perpendicular from the center of a circle to a chord bisects the chord.
   • Theorem 4: Equal chords of a circle are equidistant from the center.
   • Theorem 5: The angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle.
   • Theorem 6: Angles in the same segment of a circle are equal.

5. Angle Subtended by a Chord at a Point:

   • The angle subtended by a chord at a point on the circle is equal to the angle subtended by the same chord at any other point on the circle, provided the points lie on the same side of the chord.

6. Cyclic Quadrilateral:

   • A cyclic quadrilateral is a quadrilateral where all four vertices lie on the circumference of a circle.
   • Property: The sum of the opposite angles of a cyclic quadrilateral is 180°.

7. Tangent-Secant Theorem:

   • If a tangent and a secant (or chord) are drawn from a point outside a circle, the square of the length of the tangent is equal to the product of the lengths of the secant (or chord) segment outside the circle and the entire secant (or chord).

8. Alternate Segment Theorem:

   • The angle between the tangent and the chord through the point of contact is equal to the angle made by the chord in the alternate segment of the circle.

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Formula Sheet:

1. Length of Tangent from an External Point:

   $$\text{If } PA \text{ and } PB \text{ are tangents from point } P \text{ to the circle, then } PA = PB.$$ $$PA = \sqrt{OP^2 - r^2}$$

   where $OP$ is the distance from the external point $P$ to the center $O$ and $r$ is the radius.

2. Radius Perpendicular to Tangent:

   $$\text{If } OT \text{ is the radius at the point of contact } T \text{ of the tangent } AB, \text{ then } OT \perp AB.$$

3. Tangent-Secant Theorem:

   $$PT^2 = PA \times PB$$

   where $PT$ is the tangent from $P$ to the point $T$ on the circle, and $PA$ and $PB$ are the segments of the secant.

4. Angle Subtended by an Arc:

   $$\text{The angle subtended by an arc at the center } = 2 \times \text{ the angle subtended by the same arc on the circle.}$$

5. Cyclic Quadrilateral:

   • Opposite angles sum: $$\angle A + \angle C = 180^\circ \quad \text{and} \quad \angle B + \angle D = 180^\circ$$ where $ABCD$ is a cyclic quadrilateral.

6. Alternate Segment Theorem:

   $$\text{The angle between the tangent and the chord } = \text{Angle made by the chord in the alternate segment.}$$