One side of a triangle is divided into line segment of lenghts 6cm and 8cm by the point of tangency of the incircle of the triangle . If the radius of the incircle is 4cm , then length ( in cm ) of the longer of the two remaining sides is : ? NTSE stage 1 2013 question

Let a circle with centre O be inscribed in ∆ABC.

∴ OD = OE = OF = 4cm [ Radii of circle ]

Let BD = 6 cm and CD = 8 cm

We know that, length of two tangents drawn from an internal point to a circle are equal.

∴ BF = BD = 6 cm

CE = CD = 8 cm

Let AE = AF = *x* cm

CA = AE + CE = (*x* + 8) cm

AB = AF + BF = (*x* + 6) cm

BC = BD + CD = 6+8=14 cm

semi - perimeter of ∆ABC,

Area of ∆ABC

Also, Area of ∆ABC = Area of (∆OBC) + Area of (∆OCA) + Area of (∆OAB)

Equating (1) and (2), we get

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