\(S = \frac{a + b + c}2 =\frac{10 + 10 + x}{2} = \frac{20+x}2\)
\(S - a = \frac{20 +x}2 - 10 = \frac{20 + x - 20}2 = \frac x2\)
\(S - b = \frac{20 +x}2 - 10 = \frac{20 +x - 20}2 = \frac x2\)
\(S - c = \frac{20 + x}2 - x = \frac{20 + x- 2x}2 = \frac{20 - x}2\)
\(\therefore\) Area of isosceles triangle by heron's formula is
\(\triangle = \sqrt{S (S - a)(S - b) (S - c)}\)
\(=\sqrt{\left(\frac{20 +x}2\right) \frac x2 .\frac x2 \left(\frac{20 -x}2\right)}\)
\(= \frac x2\sqrt{\frac {400 -x^2}4} = 48\) (Given)
⇒ \(\frac {x^2} 4\times \frac{400 - x^2 } 4= 48^2 = 2304\) (By squaring on both sides)
⇒ \(400x^2 - x^4 = 2304 \times 16 = 36864\)
⇒ \(x^4 - 400x^2 + 36864 = 0\)
⇒ \(x^2 = \frac{400\pm \sqrt{160000- 147456}}2\)
\(= \frac {400\pm \sqrt {12544}}2\)
\(= \frac {400\pm 112}2\)
\(= \frac {512}2 \; or\; \frac{288}2\)
\(= 256 \; or \; 144\)
⇒ \(x = \sqrt {256} \; or\; \sqrt{144}\)
⇒ \(x = 16 \;or\; 12\)
Possible length of base is 16 cm or 12 cm.
