Question : Two circles with centres A and B touch each other externally, PQ is a direct common tangent which touches the circle at P and Q. If the radii of the circles are 9 cm and 4 cm, respectively, then the length of PQ (in cm) is equal to:
Option 1: 5
Option 2: 13
Option 3: 6.5
Option 4: 12
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Correct Answer: 12
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:
$d=9+4=13\ \mathrm{cm}$
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:
$t^2=d^2−(r{_1}−r{_2})^2$
$⇒t^2=13^2−(9−4)^2$
$⇒t^2=169−5^2$
$⇒t^2=169−25$
$⇒t^2=144$
$\therefore t=12\ \mathrm{cm}$
Hence, the correct answer is 12.
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