Equilateral Areas
Problem
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF .
Does this work for any whole number side lengths?
If not, under what circumstances does it work?
What if the lengths of the sides of the triangles had been a and b instead of 3 and 4?
Can you construct an equilateral triangle whose area is the sum of the areas of ABC and DEF? What is the new area?
Getting Started
Drawing the triangles on isometric paper and using areas bsed on triangles rather than squares might help.
Student Solutions
John wrote:
Areas of triangles using triangluar measure generate the square numbers
$1, 4, 9, 16, 25$.
So the two triangles $3$ and $4$ were a fairly special case as $3^2 + 4^2 = 5^2$
But there are others that work such as $5, 12,13$ - that is Pythagorean Triples.
In the original problem $a = 3$ and $b = 4$, so $3^2 + 4^2 = c^2$ giving $c = 5$.
This was essentially just another way of looking at Pythagoras's theorem.
In general:
The formula for the area of an equilateral triangle with side $x$ is
$\text{Area} = \frac{x^2\sqrt3}{4}$So with the two triangles with sides a and b respectively, we are looking for a third triangle with area:
$$\frac{c^2\sqrt3}{4} = \frac{a^2\sqrt3}{4} + \frac{b^2\sqrt3}{4} $$