Question

An isosceles triangle has a perimeter of 18 cm. Find a function that models its area A in terms of the length of its base b.

Answer 1

So.... notice the picture below

now, we know what the "height" or "altitude" is
now, we also know the perimeter is 18cm

so, k + k + b = 18
or 2k + b = 18
thus  $\bf k=\cfrac{18-b}{2}$

so... one can say that $\bf \textit{area of a triangle}=A=\cfrac{1}{2}bh\qquad \begin{cases} b=b \\\\ h=\sqrt{k^2-\left( \frac{b}{2} \right)^2} \\\\ h=\sqrt{k^2-\frac{b^2}{4}}\\ --------------\\ k=\frac{18-b}{2}\qquad thus\\ --------------\\ h=\sqrt{\left( \frac{18-b}{2} \right)^2-\frac{b^2}{4}} \end{cases}\\\\ -----------------------------\\\\ A=\cfrac{1}{2}\cdot b\cdot \sqrt{\left( \frac{18-b}{2} \right)^2-\frac{b^2}{4}}$

and you can simplify it if you wish.. not sure you have to