Question

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Fig. 10.14

Answer 1

A circle is tangent to sides AB and AC of triangle ABC at points E and F, respectively. The length of segment AF is denoted by x.
In triangle ABC, the following lengths are established based on tangent properties:
CF = CD = 6 cm (Tangents from point C)
BE = BD = 8 cm (Tangents from point B)
AE = AF = x (Tangents from point A)
The sides of the triangle can be expressed as:
AB = AE + EB = x + 8
BC = BD + DC = 8 + 6 = 14
CA = CF + FA = 6 + x
The semi-perimeter, s, is calculated as:
2s = AB + BC + CA
2s = (x + 8) + 14 + (6 + x)
2s = 28 + 2x
s = 14 + x
The area of triangle ABC is calculated using Heron's formula:
Area of ΔABC = \(\sqrt {s(s-a)(s-b)(s-c)}\)
= \(\sqrt {(14+x)(((14+x)-14){(14+x)-(6+x)}{(14+x)-(8+x)}}\)
= \(\sqrt {(14+x)(x)(8)(6)}\)
= \(4\sqrt {3(14x+x^2)}\)
The areas of the smaller triangles formed by the center of the circle (O) are:
Area of ∆OBC = \(\frac 12\) x OD x BC =  \(\frac 12\) x 4 x 14 = 28
Area of ΔOCA = \(\frac 12\) x OF x AC = \(\frac 12\) x 4 x (6+x) = 12+2x
Area of ΔOAB = \(\frac 12\) x OE x AB = \(\frac 12\) x 4 x (8+x) = 16+2x
The sum of the areas of these smaller triangles equals the area of triangle ABC:
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB
\(4\sqrt {3(14x+x^2)}\) = \(28 + (12+2x) + (16+2x)\)
\(4\sqrt {3(14x+x^2)}\)= \(56+4x\)
Dividing by 4 yields:
\(\sqrt {3(14x+x^2)}\) = \(14+x\)
Squaring both sides:
\(3(14x+x^2)\) = \((14+x)^2\)
\(42x+3x^2\) = \(196 + 28x + x^2\)
Rearranging into a quadratic equation:
\(2x^2+14x-196\) = \(0\)
Dividing by 2:
\(x^2+7x-98\) = \(0\)
Factoring the quadratic equation:
\(x^2+14x-7x-98\) = \(0\)
\(x(x+14) -7(x+14)\) = \(0\)
\((x+14)(x-7)\) = \(0\)
This gives two possible solutions for x:
x+14 = 0 => x = -14
x-7 = 0 => x = 7
Since the length of a side cannot be negative, x = -14 is rejected.
Therefore, x = 7.
The lengths of the sides are:
AB = x + 8 = 7 + 8 = 15 cm
CA = 6 + x = 6 + 7 = 13 cm