‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

What is the angle between a tangent and the radius at the point of contact?

\(90^\circ\) — the tangent is perpendicular to the radius at that point.

How many tangents can be drawn from an external point, and what about their lengths?

Exactly two, and they are equal in length.

What is a tangent to a circle?

A line that touches the circle at exactly one point, called the point of contact (refer to the figure in your NCERT textbook).

How many tangents can be drawn to a circle from a point inside, on, and outside it?

None from a point inside, exactly one from a point on the circle, and exactly two from a point outside the circle.

A tangent PQ touches a circle of radius 5 cm at Q, and \(OP = 13\) cm. Find PQ.

Since \(OQ \perp PQ\), triangle OQP is right-angled, so \(PQ = \sqrt{OP^2 - OQ^2} = \sqrt{169 - 25} = \sqrt{144} = 12\) cm.

What is the angle between a tangent to a circle and the radius at the point of contact?

\(90^\circ\) (they are perpendicular).

What is a secant of a circle?

A line that intersects the circle at two distinct points.

From an external point, two tangents of length 8 cm are drawn to a circle. What can you say about them?

They are equal in length (8 cm each), as tangents from an external point are always equal.

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Chapter 10: CirclesClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

Q: State the theorem about a tangent and the radius at the point of contact.

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Q: State the theorem about the lengths of two tangents from an external point.

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

Try one from this chapter

The circumference of a circle of radius r is:

Summary

A tangent to a circle is a line that touches the circle at exactly one point. This chapter proves two key results about tangents: the tangent at any point of a circle is perpendicular to the radius through the point of contact, and the lengths of the two tangents drawn from an external point to a circle are equal. These theorems are used to find unknown lengths and angles and to solve problems involving circles and the lines that touch them, deepening the study of plane geometry.

Tangent to a circleTangent perpendicular to radiusNumber and length of tangents from an external point

Key terms

Tangent

A line that touches a circle at exactly one point.

Point of contact

The single point where a tangent meets the circle.

Tangent–radius property

A tangent is perpendicular to the radius at the point of contact.

Equal tangents

Tangents from an external point have equal length.

Extra questions & answers

EXERCISE 10.1

EXERCISE 10.2

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Tangent

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A line that touches a circle at exactly one point.

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Circles