Multiple choice

One of the sides of a triangle is divided into segments of 6 and 8 units by the point of tangency of the inscribed circle. If the radius of the circle is 4, then the length of the shortest side of the triangle is :

  1. 12 units

  2. 13 units

  3. 14 units

  4. 15 units

  5. 16 units
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B Correct answer
Explanation

Let the segments be x=6, y=8, z. The sides are a=x+y=14, b=y+z, c=z+x. Semi-perimeter s = (14 + y+z + z+x)/2 = (14 + 8+z + z+6)/2 = 14+z. Area = sqrt(s(s-a)(s-b)(s-c)) = sqrt((14+z)(z)(6)(8)) = sqrt(48z(14+z)). Also Area = rs = 4(14+z). Squaring: 48z(14+z) = 16(14+z)^2. 3z = 14+z, 2z = 14, z = 7. Sides are 14, 15, 13. Shortest is 13.

AI explanation

Let the side divided into segments of 6 and 8 be the base, with the tangent lengths from the adjacent vertices being 6 and 8. Let the tangent length from the third vertex be x, making the side lengths x+6, x+8, and 14. Using Heron's formula with a semi-perimeter of x+14, the area is the square root of ((x+14)(14)(6)(8)), which simplifies to 12 times the square root of (7x plus 98). Since the inradius is 4, the area is also 4 times the semi-perimeter, giving 4x plus 56; equating and solving yields 4x plus 56 = 12 times the square root of (7x plus 98), which leads to x = 7. Therefore, the shortest side of the triangle is x+6 = 13 units.

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