Question

20. If the vertices of a triangle have their coordinates given by rational numbers. prove that the
triangle cannot be equilateral​

Answer 1

Answer:

Without loss of generality, we can choose the co-ordinates as

A=

(0,0)

B=

(x,y)

C=

(a,b)

Now, slope of

AB=

x

y

is rational slope of

AC=

a

b

is rational.

So

tan(BAC)=

⎝

⎛

1+

b x

a y

x

y

−

b

a

⎠

⎞

which is rational.

At

tan(60)=

3

, there cannot be any point with rational coefficient so angle BAC cannot be As we cannot find any rational point on the k line at 60

o

so getting an equilateral triangle is not possible.

Hence proved.