Problem
GEO-B1-M05-P024 Two Tangents and a Central Angle
#24
★★★★☆ Level 4 of 5
From point \(T\), tangents \(TA\) and \(TB\) are drawn to a circle with centre \(O\). It is known that \(\angle AOB=132^\circ\). Find \(\angle ATB\).
In quadrilateral \(AOBT\), two angles are right.
Radii to the points of tangency are perpendicular to the tangents: \(OA\perp TA\) and \(OB\perp TB\). Therefore in quadrilateral \(AOBT\), the angles at \(A\) and \(B\) are \(90^\circ\). The sum of angles in a quadrilateral is \(360^\circ\), so \(\angle ATB=360^\circ-90^\circ-90^\circ-132^\circ=48^\circ\).
Combines tangents and a central angle; a good step toward more serious problems.