CBSE Class 10 Maths Notes: Circles

  Definitions & Core Concepts

Let’s dive into the fascinating world of circles! This chapter explores key concepts related to tangents and their properties.

Circle: A closed two-dimensional figure where every point on its boundary is equidistant from a central point (the center).

Tangent: A line that touches a circle at exactly one point. This point is called the point of contact. Imagine a straight line barely grazing the edge of a circular wheel.

Secant: A line that intersects a circle at two distinct points.

Radius: A line segment joining the center of the circle to any point on the circle.

Point of Contact: The point where the tangent line touches the circle.

  Theorem 1: Tangent-Radius Relationship

Core Principle: The tangent to a circle is perpendicular to the radius at the point of contact. This is a fundamental theorem, and understanding it is crucial.

Proof: Let’s consider a circle with center O, and a tangent line AB touching the circle at point P. We need to prove that $OP \perp AB$.
Suppose OP is not perpendicular to AB. Draw a line from O to any other point Q on AB (other than P).
Since Q lies on the tangent, it must lie outside the circle.
Therefore, OQ is greater than OP (radius).
This means OP is the shortest distance from O to any point on AB.
The shortest distance from a point to a line is always the perpendicular distance.
Hence, $OP \perp AB$.

Implication: This theorem allows us to solve a variety of problems related to finding angles, lengths, and applying the Pythagorean theorem. If a problem states a line is tangent and gives the radius, you can instantly assume a right angle is formed at the point of contact.

  Formulaes & Applications (Tangent-Radius)

Formula: The Pythagorean theorem is often used in conjunction with the tangent-radius theorem. If we have a right-angled triangle formed by the radius, tangent, and a line segment from the center to a point on the tangent, then: $a^2 + b^2 = c^2$, where ‘c’ is the hypotenuse.

Example:
If the radius of a circle is 5 cm, and a tangent is drawn from a point 12 cm away from the center, what is the length of the tangent segment from that point to the point of contact?
Solution: The tangent, radius and the segment from the center to the tangent point of external location forms a right angle triangle.
Using the Pythagorean theorem: $5^2 + x^2 = 12^2$, where ‘x’ is the length of the tangent.
$25 + x^2 = 144$
$x^2 = 119$
$x = \sqrt{119}$ cm.

  Theorem 2: Equal Tangents from an External Point

Core Principle: The lengths of tangents drawn from an external point to a circle are equal.

Proof: Consider a circle with center O, and an external point P. Let PA and PB be two tangents drawn from P to the circle, touching at points A and B respectively. We need to prove that PA = PB.
Join OA, OB and OP.
$\angle OAP = \angle OBP = 90^\circ$ (Tangent is perpendicular to the radius)
In triangles OAP and OBP,
OA = OB (Radii of the same circle)
OP = OP (Common)
$\angle OAP = \angle OBP = 90^\circ$ (Proved above)
Therefore, $\triangle OAP \cong \triangle OBP$ (By RHS congruence rule)
Hence, PA = PB (By CPCT – Corresponding Parts of Congruent Triangles)

Implication: Knowing this theorem helps in solving problems where we need to find the length of a tangent or use the equality of tangent segments.

  Combining Tangents with Other Circle Properties

Approach: Many problems in this chapter will require the combination of the Tangent Theorems with other circle theorems, like the properties of chords, angles subtended by arcs, and properties of quadrilaterals, particularly cyclic quadrilaterals.

Example: A quadrilateral ABCD is drawn to circumscribe a circle. Prove that $AB + CD = AD + BC$.
Solution: Let the circle touch the sides AB, BC, CD, and DA at P, Q, R, and S respectively.
Since tangents from an external point are equal:
AP = AS
BP = BQ
CR = CQ
DR = DS
Now, $AB + CD = AP + PB + CR + RD = AS + BQ + CQ + DS = (AS + DS) + (BQ + CQ) = AD + BC$

Tips:

• Always look for right angles formed by tangents and radii.
• Identify external points and apply the equal tangents theorem.
• Consider using the Pythagorean theorem when right-angled triangles are present.
• Look for opportunities to create congruent triangles.
• Remember to use other circle properties that you have learned earlier in previous chapters.