Chords & Tangents | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Circles & Chords

What is a chord?

• A chord in a circle is any straight line that joins any two points on the circumference.
• Equal chords are equidistant from the center of the circle. This implies that if each end of a chord is connected to two radii, they will create an isosceles triangle.
  • This information is not categorized as a circle theorem, but it is immensely valuable for solving circle theorem problems.
• When tackling circle theorem questions, start by identifying any radii and mark them as equal lines. Then, examine if the radii are connected to any chords.

Circle Theorem: The perpendicular bisector of a chord is a radius

• If a radius or diameter intersects a chord in a circle, it will bisect that chord at a right angle.
• The term "bisect" means to cut something in half.
• This particular circle theorem, although less commonly seen, can be very helpful in determining equal lengths and angles. It is frequently encountered in problems involving congruent triangles.
• To identify this theorem, look for a radius and check if it intersects any chords. Typically, questions related to this theorem involve radii joined to the ends of chords, creating two congruent triangles. This visual representation is easier to comprehend than to recall from a description.

What else should I know about chords?

• Although not categorized as a circle theorem, a significant aspect to note is the formation of an isosceles triangle within a circle.
• Constructing a triangle from the center using a chord and two radii results in an isosceles triangle. This implies that at least two angles will be equal, and there will exist a line of symmetry within the triangle.
• Understanding this property is valuable in proving various circle theorems and can aid in solving geometric puzzles efficiently.

Circles & Tangents

What is a tangent?

• A tangent is a straight line outside a circle that touches the circumference only once.
• Tangents are easy to identify in circle theorem questions as they lie outside the circle distinctly.

Circle Theorem: A radius and a tangent are perpendicular

• In circle theorem questions, a tangent usually intersects a radius where it touches the circle's circumference. It's crucial to confirm the line the tangent intersects is indeed a radius.
• This theorem states that a radius and a tangent meet at a right angle (90 degrees).
• Remember, perpendicular simply means meeting at right angles. When applying this theorem in exams, emphasize the keywords 'A radius and a tangent meet at right angles.'

What else should I know about tangents?

• Although not a circle theorem, the following fact is crucial for solving some problems:
  • Two tangents drawn from the same external point to a circle are equal in length.
• This forms a kite with the two tangents meeting the circle.
• A kite consists of two congruent triangles along its main diagonal.
• The kite will include two right angles.
• The tangents end at two points: one where they meet outside the circle and the other where each touches the circle.