Question : The radius of the incircle of a triangle whose sides are 9 cm, 12 cm, and 15 cm is:
Option 1: 9 cm
Option 2: 13 cm
Option 3: 3 cm
Option 4: 6 cm
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Correct Answer: 3 cm
Solution :
Checking for the triangle by using Pythagoras theorem.
$9^2 + 12^2 = 15^2$
Thus, The triangle is right-angled.
Given, AB = 9, BC = 12 cm
OD = OE = OF = $x$ cm (radius of the circle)
And, BEOD is a square.
AD = AF = $9-x$ ($\because $ The length of the tangent from a point to the circle is the same.)
EC = CF = $12-x$ ($\because $ The length of the tangent from a point to the circle is the same.)
$\therefore$ AC = AF + FC
⇒ $9-x+12-x = 15$
⇒ $21-2x = 15$
⇒ $2x = 21-15$
⇒ $2x = 6$
⇒ $x = \frac{6}{2} = 3$
Hence, the correct answer is 3 cm.
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