Class 10 Maths Circles Notes

10.1 Introduction to Circles

A circle is a simple closed curve in a plane, where all the points on the circle are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.

• The diameter of a circle is the longest chord and is twice the length of the radius.
• A chord is a line segment joining 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.

The study of circles has a variety of applications in geometry, trigonometry, and real-world problems like wheels, gears, and circular tracks. This chapter focuses on key properties and theorems related to tangents.

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10.2 Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact.

Key Properties of Tangents:

• The tangent at any point on the circle is perpendicular to the radius at the point of contact.
• If two tangents are drawn from an external point to a circle, the tangents are equal in length.

Theorem 1: Tangent Perpendicular to Radius

If a line is tangent to a circle at a point $P$P, then the radius of the circle drawn to the point $P$P is perpendicular to the tangent.$$OP \perp PQ$$OP⊥PQ

Where $O$O is the center of the circle, $P$P is the point of contact, and $PQ$PQ is the tangent.

Example:

Consider a circle with center $O$O and a point $P$P on the circle. Draw a tangent at $P$P which touches the circle at $P$P. The line segment $OP$OP (radius) will be perpendicular to the tangent at point $P$P.

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10.3 Number of Tangents from a Point on a Circle

One of the interesting properties of tangents is that we can draw two tangents from any external point to a circle.

Theorem: Number of Tangents from an External Point

From an external point $A$A, we can draw exactly two tangents to a circle. These tangents are equal in length and touch the circle at two distinct points.

• The tangent-secant theorem or two tangent theorem states that the tangents drawn from an external point to a circle are of equal length.

$$PA = PB$$PA=PB

Where $P$P and $B$B are the points of contact of the tangents, and $A$A is the external point.

Example:

If we have an external point $A$A and a circle with center $O$O, we can draw two tangents $PA$PA and $PB$PB from point $A$A to the circle. The lengths of these tangents are equal.

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10.4 Summary

In this chapter, we studied the following important concepts related to circles:

1. Introduction to Circles: We defined a circle, its radius, diameter, and key terms like chord, secant, and tangent.
2. Tangent to a Circle: A tangent touches the circle at exactly one point. The tangent is perpendicular to the radius at the point of contact.
3. Number of Tangents from an External Point: From an external point, we can draw exactly two tangents to a circle, and they are of equal length.

Understanding the properties of tangents is important for solving problems in geometry and is foundational for studying more complex topics like circles in trigonometry and coordinate geometry.

MCQs Based on the “Circles” Chapter:

1. A tangent to a circle is always:

a) Parallel to the radius
b) Perpendicular to the radius
c) Equal to the diameter
d) None of the above

Answer: b) Perpendicular to the radius

2. From an external point, how many tangents can be drawn to a circle?

a) 1
b) 2
c) 3
d) 4

Answer: b) 2

3. The length of the tangents drawn from an external point to a circle is:

a) Always zero
b) Always equal
c) Always greater than the radius
d) Always less than the radius

Answer: b) Always equal

4. If OPOPOP is the radius and PQPQPQ is the tangent, then the angle between OPOPOP and PQPQPQ is:

a) $0^\circ$0∘
b) $45^\circ$45∘
c) $90^\circ$90∘
d) $180^\circ$180∘

Answer: c) $90^\circ$90∘

5. If a point PPP lies on the circumference of a circle and a line is drawn from PPP to any external point, how many tangents can be drawn from the external point to the circle?

a) 1
b) 2
c) 3
d) 0

Answer: b) 2