Question

O is the centre of the incircle of the right triangle ABC, as shown. OQBR is of what shape?

Answer 1

Observe that quadrilateral OQBR has $\angle QBR={90}^{\circ }$

Since AB and BC are tangents to the circle, OQ and OR are perpendicular to AB and BC respectively and so $\angle BQOand\angle BRO$ are also ${90}^{\circ }$.

Since three angles in a quadrilateral are ${90}^{\circ }$, the fourth angle will be ${360}^{\circ }$ - 3( ${90}^{\circ }$) = ${90}^{\circ }$. So it is a rectangle.

Also, since $OQ=OR$, it follows that all the sides are equal and hence quadrilateral OQBR is actually a square.