Let the given side be $5$;
My approach to solving this is:
let $\phi$ be the angle formed by the side and height of the triangle, and $θ$ be the angle between the base and side.
then,
$$2θ + 2\phi = 180$$
or,
$$ \sin^{−1}(h/5) + \sin^{−1}\Big(\frac{b/2}{h}\Big) = \sin^{−1}(1)$$
or,
$$h/5 + b/2h = 1 $$
or,
$$2h^2 + 5b = 10h \hspace{1cm} \text{—- eqn (1)}$$
Using Pythagorean theorem:
$$h^2 + (b/2)^2 = 5^2 \hspace{1cm} \text{—- eqn (2)}$$
Now, solving these two equations and getting h and b doesn't seem good. So how should I find out the $h$ and $b$ when I am given with one side of the isosceles triangle?
Best Answer
You can't solve it based only on the side given. The problem is missing information.
If the side is $c$, pick any length $a$, $0\lt a\lt 2c$ - and you can construct an isosceles triangle with base $a$ and side $c$.
Once the base is fixed, the triangle is uniquely determined up to congruency, and the height corresponding to the base is $h=\sqrt{c^2-\left(\frac{a}{2}\right)^2}$