ABC is a right triangle, right angled at B. A circle is inscribed in it. The lengths of the two sides containing the right angle are 6 cm and 8 cm. Find the radius of the incircle.
Let the radius of the in circle be x cm.
Let the in circle touch the side AB, BC and CA at D, E, F respectively.
Let O be the centre of the circle.
Then, OD = OE = OF = x cm.
Also, AB = 8 cm and BC = 6 cm.
Since the tangents to a circle from an external point are equal, we have
AF = AD = (8 – x) cm, and
CF = CE = (6 – x) cm.
∴ AC = AF + CF = (8 – x) cm + (6 – x) cm
= (14 – 2x) cm.
`AC^2 = AB^2 + BC^2`
`⇒ (14 – 2x)^2 = 8^2 + 6^2 = 100 = (10)^2`
⇒ 14 – 2x = ±10 ⇒ x = 2 or x = 12
⇒ x = 2 [neglecting x = 12].
Hence, the radius of the in circle is 2 cm.
Concept: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
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