Question : $\triangle$ ABC is a right-angled triangle with AB = 6 cm, BC = 8 cm. O is the incentre of the triangle. The radius of the in-circle is:
Option 1: 3 cm
Option 2: 4 cm
Option 3: 2 cm
Option 4: 5 cm
Correct Answer: 2 cm
Solution :
AB = 6 cm, BC = 8 cm
By Pythagoras theorem,
$AC^2=AB^2+BC^2$
$AC = \sqrt{6^2+8^2}$ = $\sqrt{36+64}$ = $\sqrt{100}$ = $10$ cm
$OP = OQ = OR = r$
Since tangents drawn from a point to the circle are equal,
$RB = BP = r$
$PA = AQ = 6-r$
$RC = CQ = 8-r$
$AC = AQ + QC$
$10 = 6-r + 8-r$
⇒ $2r = 14 – 10 = 4$
⇒ $r = 2$ cm
Hence, the correct answer is 2 cm.
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