★ AIMS Think+ · Q8

Given:

Origin O$\left(0,0\right)$.

To Find:

Valid examples of the requested triangles.

Property:

Perpendicular axes give a right angle; reflection gives equal lengths.

Solution:

For a right-angled isosceles triangle, choose A$\left(2,0\right)$, B$\left(0,2\right)$. OA = OB = 2 and ∠AOB = 90°.

For the second triangle, choose U$\left(-2,-3\right)$ in III and V$\left(2,-3\right)$ in IV. ${\mathrm{OU}}^2$ = ${\mathrm{OV}}^2$ = 13.

Results:

OAB and OUV satisfy the respective requirements. Many other constructions are valid.