Q148 / 200 · AP EAMCET 2002 · Mathematics

The perimeter of a triangle is $16 \mathrm{~cm}$, one of the sides is of length $6 \mathrm{~cm}$. If the area of the triangle is $12 \mathrm{sq} \mathrm{cm}$. Then the triangle is

  • A
    right angled
  • Correct answer
    B
    isosceles
  • C
    equilateral
  • D
    scalene

Solution

Given that $$ \begin{aligned} & 2 s=16 \Rightarrow s=8 \mathrm{~cm} \\ & 2 s=a+b+c, a=6 \mathrm{~cm}, \Delta=12 \\ & \Delta^2=s(s-a)(s-b)(s-c) \\ & 144=8(8-6)(8-b)(8-c) \\ & {\left[\begin{array}{r} \because a+b+c=16 \\ b+c=10 \\ b=10-c \end{array}\right]} \\ & 18=2(8-b)(8-c) \\ & 18=2(8-10+c)(8-c) \\ & 9=(-2+c)(8-c) \\ & 9=-16+10 c-c^2 \Rightarrow c^2-10 c+25=0 \\ & \Rightarrow \quad(c-5)^2=0 \\ & \Rightarrow \quad c=5 \\ & \end{aligned} $$ Then, $b=10-5=5 \Rightarrow b=c$ Therefore, triangle is isosceles.

Hence, the correct answer is option: 'B'.

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