Explain why the following statement is true: If the perpendicular distance of a chord from the centre Class 9

Explain why the following statement is true: If the perpendicular distance of a chord from the centre Class 9

Question 1.

Explain why the following statement is true: If the perpendicular distance of a chord from the centre is d and the radius is r, then the chord length is $2 \sqrt{r^2-d^2}$. Class 9

Answer:

Given : Radius of circle = r

Perpendicular distance from centre to chord = d

To prove : Chord length = $2 \sqrt{r^2-d^2}$.

Proof

The perpendicular from the centre to a chord bisects the chord.

∴ MB = $\frac{\mathrm{AB}}{2}$

In right-angled ∆OMB,

OB² = OM² + BM² ...[By Baudhayana-Pythagoras theorem]

∴ r² = d² + BM²

∴ BM² = r² - d²

∴ BM = $\sqrt{r^2-d^2}$

∴ AB = 2 × BM = $2 \sqrt{r^2-d^2}$

∴ Chord length = $2 \sqrt{r^2-d^2}$