Question

Explain why it is not possible to construct an equilateral triangle that has threevertices with integer coordinates. Show your work by providing an example andincluding a graph. Consider an equilateral triangle whose base lies along the x-axis.

Answer 1

Step-by-step explanation:

Suppose we place a triangle on a graph as shown below.

Let us also suppose that a is an integer. Meaning the bases of our equilateral triangle have integer coordinates.

Now the question is, what is the coordinate of the upper vertex, the top of the triangle?

Well, the Pythagoras's theorem tells us that

`h^2+((a)/(2))^2=a^2`

which can also be written as

`h^2+(a^2)/(4)=a^2`

subtracting a^2/4 from both sides gives

`h^2+(a^(2))/(4)-(a^(2))/(4)=a^2-(a^(2))/(4)`
`\Rightarrow h^2=a^2-(a^(2))/(4)`
`\Rightarrow h^2=(4a^2)/(4)-(a^2)/(4)`
`h^2=(3a^2)/(4)`

taking the square root of both sides gives

`\boxed{h=(√(3))/(2)a.}`

This means that if a is an integer, h cannot be an integer! Why? Because √3 is never an integer!

Since cannot be an integer, this means that if we place the bases of our equilateral triangle at integer coordinates then the top of the triangle will NOT have integer coordinates!