The line from the centre that cuts a chord in half
Show on a big screenA stick lies flat across a round pan. How far from the centre is it?
- A circle, a chord, and the centre O.
- A line from O, meeting the chord at 90°.
- Two equal marks on the two halves of the chord.
- Two radii, one to each end of the chord.
The rule
The distance is the length of the 90° line from the centre.
Take a chord that is not a diameter. A 90° line from the centre meets it at its middle. Join the centre to both ends of the chord. This makes two triangles. The two radii are equal. The 90° line is shared. Both triangles have a right angle. So they match: right angle, equal radii, shared line. Matching triangles have equal sides, so the two halves are equal.
The pan, with numbers
The pan's radius is 17 cm. A stick 30 cm long has halves of 15 cm each. The radius, one half and the distance form a right triangle. The radius is its longest side. So distance² = 17² − 15². Here 17² = 289 and 15² = 225. Take away: 289 − 225 = 64. The square root of 64 is 8. The stick sits 8 cm from the centre.
| Stick | Half | 289 minus half squared | Distance out |
|---|---|---|---|
| 34 cm | 17 cm | 289 − 289 = 0 | 0 cm |
| 30 cm | 15 cm | 289 − 225 = 64 | 8 cm |
| 16 cm | 8 cm | 289 − 64 = 225 | 15 cm |
It works the other way too. Draw a line from a chord's middle to the centre. That line meets the chord at 90°. Why? The two triangles match, by three equal sides. So the angles at the middle are equal. They fill a straight line, so each is 90°. A diameter has no such line, because its middle is the centre.
Slide the chord away from the centre. Read the two halves.
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Try the lab to finish this lesson.