Review question

# Can we find this circle's integer and non-integer rational points? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9072

## Suggestion

1. The point with coordinates $(a,b)$, where $a$ and $b$ are rational numbers, is called:

• an integer rational point if both $a$ and $b$ are integers;

• a non-integer rational point if neither $a$ nor $b$ is an integer.

1. Write down an integer rational point and a non-integer rational point on the circle $x^2+y^2=1$.

What are the simplest possible points here? Where does the circle cut the axes?

Could we see a point on the unit circle as a vertex of a right-angled triangle, with the circle’s radius as the hypotenuse?

What right-angled triangles do we know with integer-length sides? Can we use these?

1. Write down an integer rational point on the circle $x^2+y^2=2$. Simplify $(\cos\theta+\sqrt{m}\sin\theta)^2+(\sin\theta-\sqrt{m}\cos\theta)^2$ and hence obtain a non-integer rational point on the circle $x^2+y^2=2$.

What must the value of $m$ be?

What values of $\cos\theta$ and $\sin\theta$ did we use in part (a) in our Pythagorean triple right-angled triangle?