Question : In the following figure, the circles with centres B and D have radii of 4 cm and $x$ cm, respectively. AC is a tangent to both circles. Find the value of $x$.
Option 1: 6 cm
Option 2: 3 cm
Option 3: 7 cm
Option 4: 5 cm
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Correct Answer: 6 cm
Solution :
Here, $\angle$DAC = $\angle$ACB = 90° (Tangnet is perpendicular to the radius at the point of contact)
$\angle$AED = $\angle$BEC (Vertically opposite angles are equal)
By AA similarity criteria, $\triangle$ AED $\sim\triangle$ CEB
$\frac{AE}{EC} = \frac{AD}{BC}$
⇒ $\frac{6}{9}=\frac{4}{x}$
$\therefore x=\frac{4\times9}{6} = 2\times 3 = 6 \ \mathrm{cm}$
Hence, the correct answer is 6 cm.
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