Question

In Figure, if $ OA=5 cm$, $ AB=8 cm$ and $ OD$ is perpendicular to $ AB$, then $ CD$ is equal to:

(4)

• A. $2cm$
• B. $3cm$
• C. $4cm$
• D. $5cm$

Answer 1

The correct option is A

$2cm$

Step 1: We know that

Radius of the circle $=r=AO=5cm$

Length of chord AB $=8cm$

Since the line is drawn through the center of a circle to bisect a chord is perpendicular to the chord

$AOC$ is a right-angled triangle with $C$ as is drawn

The radius the bisector of $AB$.

∴ $AC=\frac{1}{2}\left(AB\right)=\frac{8}{2}=4cm$

Step 2: Apply Pythagoras theorem.

On applying Pythagoras theorem to right-angled triangle $AOC$,

${\left(AO\right)}^2={\left(OC\right)}^2+{\left(AC\right)}^2$

$\Rightarrow {\left(5\right)}^2={\left(OC\right)}^2+{\left(4\right)}^2$

$\Rightarrow {\left(OC\right)}^2={\left(5\right)}^2-{\left(4\right)}^2$

$\Rightarrow {\left(OC\right)}^2=25-16$

$\Rightarrow {\left(OC\right)}^2=9$

Step 3: Taking square root on both sides.

$\Rightarrow$ $\left(OC\right)=3$

∴ The distance of from the center of the circle is$3cm$.

Now, $OD$ is the radius of the circle,

∴ $OD=5cm$

$CD=OD-OC$

$CD=5-3$

$CD=2$

Therefore,$CD=2cm$.

Hence, the correct option is option(A).