Chords, Pythagoras and Inscribed Figures ഞാൺ
Many questions in this chapter give you a chord, a square or a triangle inside a circle and ask for the circle's size. The key tools are two: the perpendicular from the centre cuts a chord in half, and Pythagoras' theorem.
- A chord (ഞാൺ) joins two points of a circle; the diameter is the longest chord.
- Pythagoras: in a right triangle, (hypotenuse)² = (one side)² + (other side)².
- A regular hexagon in a circle has side = radius.
1. The perpendicular from the centre halves a chord
If a chord is at distance d from the centre (measured along the perpendicular) and has length c, then r² = d² + (c/2)².
2. Chord lab
Try half-chord 3 and distance 4: radius 5. Half-chord 2 and distance 3: r² = 13.
3. Squares, rectangles and hexagons inside circles
When all four corners of a square or rectangle lie on a circle, each diagonal passes through the centre (the corner angles are right angles), so diagonal = diameter. For a regular hexagon, side = radius, and the longest diagonal = diameter.
4. A circle through the corners of an isosceles triangle
5. Area of an equilateral triangle
An archaeologist who finds a broken piece of a round plate measures a chord and its distance from the arc's middle, and works out the plate's original radius with exactly this rule. Engineers use it to size circular arches and bridges.
6. Check yourself
- The perpendicular from the centre to a chord halves it: r² = d² + (half chord)².
- Square or rectangle in a circle: diagonal = diameter. Regular hexagon: side = radius.
- Centre of the circle through an isosceles triangle lies on its height; find it with Pythagoras.
- Equilateral triangle of side a: height (√3/2)a, area (√3/4)a².