Perpendicular Tangents

Geometry Level 2

$A$ and $B$ are points on a circle such that their tangents are perpendicular (at $T$ ).

If $C$ is on the major arc $AB$ , then what is $\angle ACB$ ?

60 ^ \circ

90 ^ \circ

30 ^ \circ

45 ^ \circ

1 solution

Ayush G Rai
Oct 28, 2016

Let the center of the circle be $O.$ The quadrilateral $TAOB$ is a square since $OA=OB,\angle OAT=\angle OBT=90^\circ.$ So $\angle BOA=90^\circ.$
Therefore $\angle ACB=\dfrac{\angle BOA}{2}=\dfrac{90^\circ}{2}=\boxed{45^\circ}.$ [Angle at the circumference is half the angle at the center].

Perpendicular Tangents

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