Changing Triangles

The purple triangle has a height of 5 units and a width of 1 unit which gives it an area of $\frac{1}{2}\times5\times1 = 2.5$ units $^2$ .

The yellow triangle has a height of 4 units and a width of 2 units which gives it an area of $\frac{1}{2}\times4\times2 = 4$ units $^2$ .

The pink triangle has a height of 3 units and a width of 3 units which gives it an area of $\frac{1}{2}\times3\times3 = 4.5$ units $^2$ .

Therefore the pink triangle has the largest area

The formula for the area of the triangle is $\dfrac{xy}{2}$ ,where $x$ and $y$ are the length of the base and height of the triangle respectively.Since $x,y>0$ ,we can apply the Arithmetic Mean-Geometric Mean Inequaltiy for two variables,which states that $\dfrac{x+y}{2}\geq \sqrt{xy}\;\forall\; x,y>0$ .

From this we get that $\dfrac{xy}{2}\leq \dfrac{(x+y)^2}{8}$ .Now,for the AM-GM Inequality,equality occurs only when all the variables are equal i.e $x=y$ (and that will be the maximum possible area of the triangle,by the AM-GM Inequality).

Only the red triangle has both its base and height equal.Therefore,it has the largest area,which is $\dfrac{xy}{2}=\dfrac{3(3)}{2}=\color{#D61F06}{\boxed{4.5}}$

If we look at the triangles' areas in terms of units of squares, then we can see that the pink triangle is the largest in area.

First, we must consider the formula for the area of a triangle:

$\frac{B*H}{2}$

Where B is the base of the triangle and H is the height. We can then input the values of the triangles' dimensions in units of squares. Since all of the triangles are being measured in squares, it doesn't matter that it isn't a formal unit.

We can calculate the purple triangle to be:

$\frac{B*H}{2}$ $=$ $\frac{1*5}{2}$ $= 2.5$ squares $^2$

We can use the same calculation, but input the different dimensions for the other triangles:

Yellow: $\frac{B*H}{2}$ $=$ $\frac{2*4}{2}$ $= 4$ squares $^2$

Pink: $\frac{B*H}{2}$ $=$ $\frac{3*3}{2}$ $= 4.5$ squares $^2$

We can then see that the pink triangle is the largest, the area being $4.5$ squares $^2$