Problem of the Week

Problem 5

A triangle with sides of length $a$ , $b$ and $c$ has area $A$ .

(i) By considering the angle $\theta$ between the sides of lengths $a$ and $b$ , or otherwise, show that $16A^2 = 4a^2b^2 - (a^2 + b^2 - c^2)^2$ . (You may use the identity $(\sin \theta)^2 + (\cos \theta)^2 = 1$ .)

(ii) Deduce that $A = \sqrt{s(s-a)(s-b)(s-c)}$ , where $s$ is the semiperimeter $\frac{a+b+c}{2}$ .

(iii) Hence show an equilateral triangle has a greater area than any other triangle of the same perimeter.

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Problem of the Week

Problem 5

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