# NCERT Class 10 Mathematics Chapter 10 Circles: Summary, Concepts, and Revision Notes | SwaVid This chapter introduces the fundamental concepts of circles, building upon your prior knowledge. It focuses specifically on the relationship between a c... Canonical: https://swavid.com/maths/class/10/chapter/circles Source: https://swavid.com/maths/class/10/chapter/circles # Circles ## Concepts in this chapter ## The ideas this chapter keeps returning to ## Where this chapter usually trips people up ## Find the gaps before this chapter finds them for you. ## Related chapters ### Theorem 10.1: Perpendicularity of Radius and Tangent Chapter 10 · Class 10 Mathematics This chapter introduces the fundamental concepts of circles, building upon your prior knowledge. It focuses specifically on the relationship between a circle and a line, particularly tangents. Understanding these concepts is crucial for further studies in geometry and for solving various real-world problems involving circular shapes. Your study route Key topics Theorem 10.1 states: 'The tangent at any point of a circle is perpendicular to the radius through the point of contact.' This means the angle formed between the radius and the tangent at their intersection point is always 90 degrees. Example Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Watch out Assuming the tangent is perpendicular to any line from the center, not specifically the radius through the point of contact. Forgetting that this applies only at the point of contact. Theorem 10.1 states: 'The tangent at any point of a circle is perpendicular to the radius through the point of contact.' This means the angle formed between the radius and the tangent at their intersection point is always 90 degrees. Tap the card for an example Example Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Why it matters This is a fundamental property used extensively in solving problems involving tangents, finding lengths, and proving other geometric relationships. It often forms a right-angled triangle, allowing the use of the Pythagorean theorem. Watch out Assuming the tangent is perpendicular to any line from the center, not specifically the radius through the point of contact. Forgetting that this applies only at the point of contact. Ask at home Ask the child to draw a circle, a radius, and a tangent at the end of that radius. Then ask them to explain the angle formed and why it's important. Provide a simple problem where this theorem is needed to find a missing length. Chapter 10, 'Circles', delves into the properties of tangents to a circle. It begins by defining a tangent as a line that intersects the circle at exactly one point, called the point of contact, distinguishing it from a secant which intersects at two points. A key theorem established is that the tangent at any point of a circle is perpendicular to the radius through the point of contact. The chapter then explores the number of tangents that can be drawn to a circle from a point, categorizing them into three cases: zero tangents from an interior point, one tangent from a point on the circle, and exactly two tangents from an external point. Another significant theorem states that the lengths of tangents drawn from an external point to a circle are equal. These theorems form the basis for solving various geometric problems related to circles and tangents. Chapter summary Chapter 10, 'Circles', delves into the properties of tangents to a circle. It begins by defining a tangent as a line that intersects the circle at exactly one point, called the point of contact, distinguishing it from a secant which intersects at two points. A key theorem established is that the tangent at any point of a circle is perpendicular to the radius through the point of contact. The chapter then explores the number of tangents that can be drawn to a circle from a point, categorizing them into three cases: zero tangents from an interior point, one tangent from a point on the circle, and exactly two tangents from an external point. Another significant theorem states that the lengths of tangents drawn from an external point to a circle are equal. These theorems form the basis for solving various geometric problems related to circles and tangents. What you should learn Keep these close A tangent intersects a circle at exactly one point (point of contact). A secant intersects a circle at two distinct points. The radius drawn to the point of contact is perpendicular to the tangent (Theorem 10.1). The angle between the radius and the tangent at the point of contact is 90 degrees. No tangent can be drawn from a point inside the circle. Exactly one tangent can be drawn from a point on the circle. Exactly two tangents can be drawn from a point outside the circle. The lengths of the two tangents drawn from an external point to a circle are equal (Theorem 10.2). The line joining the center to the external point bisects the angle between the two tangents. The line joining the center to the external point bisects the angle between the radii to the points of contact. The quadrilateral formed by the two radii, the two tangents, and the external point has two right angles. Pythagorean theorem is often used in problems involving tangents and radii. Common confusions It is easy to think A tangent is just any line that touches the circle. The clearer idea A tangent must intersect the circle at exactly one point. The defining property is the single point of intersection. It is easy to think The radius is perpendicular to the tangent at any point on the tangent. The clearer idea The radius is perpendicular to the tangent only at the point of contact. At any other point on the tangent, the line segment from the center to that point is longer than the radius and is not perpendicular to the tangent. It is easy to think The two tangents from an external point are always parallel. The clearer idea The two tangents from an external point are never parallel; they meet at the external point. Their lengths are equal, but they diverge from the external point. It is easy to think Confusing a tangent with a chord or a secant. The clearer idea A tangent touches at one point. A chord connects two points on the circle. A secant is a line that passes through two points on the circle. They are distinct concepts. Before you push ahead Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns. Keep exploring Class 10 Source - Define and identify tangents and secants of a circle. - State and apply Theorem 10.1 regarding the perpendicularity of a radius and a tangent at the point of contact. - Determine the number of tangents that can be drawn to a circle from a given point (inside, on, or outside). - State and apply Theorem 10.2 regarding the equality of lengths of tangents drawn from an external point. - Solve problems involving properties of tangents to a circle. - Define and identify tangents and secants of a circle. - State and apply Theorem 10.1 regarding the perpendicularity of a radius and a tangent at the point of contact. - Determine the number of tangents that can be drawn to a circle from a given point (inside, on, or outside). - State and apply Theorem 10.2 regarding the equality of lengths of tangents drawn from an external point. - Solve problems involving properties of tangents to a circle. - NCERT Class 10 Mathematics textbook: Mathematics : Chapter 10: Circles ## Key Links - [NCERT Mathematics](https://swavid.com/maths) - [Class 10](https://swavid.com/maths/class/10) - [Start with concepts](https://swavid.com/maths/class/10/chapter/circles) - [Check a prerequisite](https://swavid.com/learning-debt-identifier) - [Concepts](https://swavid.com/maths/class/10/chapter/circles) - [Practice](https://swavid.com/maths/class/10/chapter/circles/practice-questions) - [NCERT Solutions](https://swavid.com/maths/class/10/chapter/circles/ncert-solutions) - [Find the gaps before this chapter](https://swavid.com/learning-debt-identifier) - [Learn this chapter adapted to you](https://swavid.com/learning-style-test) - [Previous chapter 9 . 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