Question

State the following statement is true or false

The measure of the line segment joining the centre of a circle to the mid-point of a chord is $\frac{\sqrt{4r^2-L^2}}{2}$

• A. True
• B. False

Answer 1

The correct option is A True

Suppose AB is the chord. O is the centre of the circle. If
OM is the perpendicular drawn from O to AB as its distance, then OAM and OMB

are both right angled triangles.
Since, length of the chord is L, both AM and MB $=\frac{L}{2}$
If r is the radius of the circle, then
$=>{OB}^2={OM}^2+{MB}^2$
$=>r^2={OM}^2+{\left(\frac{L}{2}\right)}^2$
$=>{OM}^2=r^2-\frac{L^2}{4}$
$=>{OM}^2=\frac{4r^2-L^2}{4}$
=> Distance $=OM=\frac{\sqrt{4r^2-L^2}}{2}$