Grade 10 Math Question

Let's say the sides are $a,b$ and $c$ , where $c$ is the longest side.

Assume the 2 unknown angles are $x$ and $y$ .

By Trigonometry Formula, we obtain:

$sin x=$ $\frac{a}{c}$

$cos x=$ $\frac{b}{c}$

[SIMPLIFY] $a = c sinx$ and $b=c cosx$

Perimeter $P$ = $a + b + c$

[SUBSTITUTE] $P = csinx + ccosx + c$

$P' = ccosx - csinx$ [DIFFERENTIATE]

At maximum, $P' = 0$ :

$0 = ccosx - csinx$

$ccosx = csinx$

$\frac{sinx}{cosx}$ $= 1$

$tanx = 1$ , where $x < 90 degrees$

$x = 45 degrees$ , then $y = 180 - 45 - 90 = 45 degrees$

We get $90 degrees, 45 degrees, 45 degrees$ -> Isosceles right-angled triangle

[PROVEN]