Q1.
Activity: Draw a circle. Draw chords of various lengths. Drop a perpendicular to each chord from the centre. Record the length of the chord and its distance from the centre in a table. (Table 1: Length of Chord / Distance from Centre.) What do you observe?
Answer
Observation: the longer the chord, the smaller its distance from the centre.
Here is the table filled in for a circle of radius 5 cm (do the same for your own circle and compare).
| Length of chord (cm) | 10 | 9 | 8 | 6 | 4 | 2 |
|---|---|---|---|---|---|---|
| Distance from centre (cm) | 0 | 2.18 | 3 | 4 | 4.58 | 4.90 |
Each entry comes from the same right triangle:
d = √(r² − (½ chord)²) = √(25 − (½ chord)²)
e.g. chord 8 ⇒ d = √(25 − 16) = 3; chord 6 ⇒ d = √(25 − 9) = 4
d = √(r² − (½ chord)²) = √(25 − (½ chord)²)
e.g. chord 8 ⇒ d = √(25 − 16) = 3; chord 6 ⇒ d = √(25 − 9) = 4
Why it happens: r is fixed, so d² + (½ chord)² is a constant (= 25 here). If one of the two squares grows the other must shrink. So a longer chord forces a smaller distance — which is exactly Theorem 8.
Check the two extremes: the longest chord is the diameter, 10 cm, at distance 0 — it passes through the centre. Push the chord outwards and it shrinks towards a single point, with distance approaching the full radius, 5 cm.