Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
Circles - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Circles aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define a circle and its key components.
A circle is a collection of all points in a plane at a constant distance (radius) from a fixed point (centre). Key components include chord, segment, sector, and arc.
Understand the three positions of a line relative to a circle.
A line can be non-intersecting (no common points), a secant (two common points), or a tangent (one common point) to a circle.
Define tangent and its unique property.
A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact, and is perpendicular to the radius at that point.
Theorem: Tangent is perpendicular to 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. This is fundamental for solving problems involving tangents.
Number of tangents from a point to a circle.
From a point inside the circle: no tangent; on the circle: one tangent; outside the circle: two tangents.
Theorem: Lengths of tangents from an external point are equal.
If two tangents are drawn from an external point to a circle, their lengths are equal. This is useful in geometric proofs and constructions.
Real-world application of tangents.
Tangents are seen in real life, like the rope of a pulley system acting as tangents to the pulley wheel, demonstrating the practical importance of the concept.
Activity to visualize tangents.
Using a circular wire and a straight wire, one can visualize how a tangent is formed when the straight wire touches the circle at exactly one point.
Angle between two tangents from an external point.
The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
Example: Prove tangents from an external point are equal.
By joining the external point to the centre and using right triangles, one can prove that the lengths of the two tangents are equal.
Example: Find the length of a tangent.
Given a point outside the circle and the radius, the length of the tangent can be found using the Pythagorean theorem.
Chord properties related to tangents.
The chord of a larger circle that touches a smaller concentric circle is bisected at the point of contact, illustrating the relationship between chords and tangents.
Parallel tangents and their properties.
A circle can have at most two parallel tangents, which are equidistant from the centre and lie on opposite sides of the circle.
Quadrilateral circumscribing a circle.
If a quadrilateral circumscribes a circle, the sum of the lengths of opposite sides are equal (AB + CD = AD + BC).
Triangle circumscribing a circle.
A triangle drawn to circumscribe a circle divides the sides into segments whose lengths relate to the triangle's sides and the circle's radius.
Misconception: More than two tangents from an external point.
A common misconception is that more than two tangents can be drawn from an external point to a circle, but the maximum is always two.
Memory hack: Tangent properties.
Remember 'One radius, one tangent' at the point of contact, and 'Equal lengths' for tangents from an external point.
Diagram: Understanding tangent and secant.
Drawing diagrams helps visualize the difference between a tangent (touching at one point) and a secant (intersecting at two points).
Theorem: Angle between tangent and chord.
The angle between a tangent and a chord is equal to the angle in the alternate segment, a useful property for solving circle geometry problems.
Summary of circle theorems.
Key theorems include: tangent perpendicular to radius, equal lengths of tangents from an external point, and angles in the same segment are equal.
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