Question : If two circles of radii 18 cm and 8 cm touch externally, then the length of a direct common tangent is:
Option 1: 24 cm
Option 2: 14 cm
Option 3: 16 cm
Option 4: 12 cm
Correct Answer: 24 cm
Solution :
Let $r{_1}$ and $r{_2}$ be the radii of the two circles, and let $d$ be the distance between their centres. In this case:
$d=r{_1}+r{_2}$
Given that the radii are 18 cm and 8 cm, we have:
$d=18+8=26$ cm
Now, the direct common tangent is the line segment that joins the points of contact of the two circles. This forms a right-angled triangle with the line segment connecting the centres. The length of the direct common tangent can be found using the Pythagorean theorem.
Let $t$ be the length of the direct common tangent. Then:
$t^2=d^2−(r{_1}−r{_2})^2$
⇒ $t^2=26^2−(18−8)^2$
⇒ $t^2=676−10^2$
⇒ $t^2=676−100$
⇒ $t^2=576$
⇒ $t=24$ cm
Hence, the correct answer is 24 cm.
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