Question

The area of an isosceles triangle having equal side 5 cm is 12 cm${}^2$. The third side of the triangle is

• A. 8 cm
• B. 4 cm
• C. 7 cm
• D. 10 cm

Answer 1

The correct option is A 8 cm

Let the equal sides of an isosceles triangle be a and un-equal side be b.
$\therefore a=5cm$
Now, area of an isosceles triangle is $\frac{b}{4}\sqrt{4a^2-b^2}$
$\Rightarrow 12=\frac{b}{4}\sqrt{4(5{)}^2-b^2}$
$\Rightarrow 48=b\sqrt{100-b^2}$
$\Rightarrow 2304=b^2(100–b^2)$
$\Rightarrow b^4–100b^2+2304=0$
$\Rightarrow b^4–64b^2–36b^2+2304=0$
$\Rightarrow b^2(b^2–64)–36(b^2–64)=0$
$\Rightarrow (b^2–64)(b^2–36)=0$
$\Rightarrow b^2=64$ or $b^2=36$
$\Rightarrow b=\pm 8$ or $b=\pm 6$
$\Rightarrow b=8$ or $b=6$ (Neglecting negative values) Therefore, the third side of the triangle can be 8 cm or 6 cm.
Hence, the correct answer is option (a).