Question

At one end $A$of a diameter $AB$ of a circle of radius $5cm$, tangent $XAY$ is drawn to the circle. The length of the chord $CD$ parallel to $XY$ and at a distance $8cm$ from $A$ is:

• A. $4cm$
• B. $5cm$
• C. $6cm$
• D. $8cm$

Answer 1

The correct option is D

$8cm$

Step 1. According to the given details

Radius of circle, $AO=OC=5cm$

$AE=8cm$

$AE=OE+AO$

$OE=AE-AO$

$OE=8-5=3cm$

Step 2. As a line drawn from centre of the circle to the chord is perpendicular to the chord

Therefore $OE$ is perpendicular to the chord $CD$.

In $∆OEC$, $\angle OEC=90°$

Step 3. Using Pythagoras theorem,

$$\begin{gathered} O{C}^2=O{E}^2+E{C}^2 \\ \Rightarrow E{C}^2=O{C}^2-O{E}^2 \\ \Rightarrow E{C}^2=5^2-3^2 \\ \Rightarrow E{C}^2=25-9 \\ \Rightarrow =16 \\ \Rightarrow EC=4cm \end{gathered}$$

Step 4. As a line drawn from center of the circle to the chord is bisect the chord

Therefore,

$$\begin{gathered} CD=2xEC \\ =2x4 \\ =8cm \end{gathered}$$

Hence option (D) is the correct option.