This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the properties of tangents and radii. Since RS and RT are tangent lines to the circle at points S and T respectively, and O is the center of the circle, the radii OS and OT are perpendicular to the tangents RS and RT. Therefore, we have: m( OSR) = 90^ m( OTR) = 90^ Step 2: Find the measure of the central angle SOT. Consider the quadrilateral OSRT. The sum of the interior angles of a quadrilateral is 360^. We are given m( SRT) = 40^. So, we can write: m( SOT) + m( OSR) + m( OTR) + m( SRT) = 360^ Substitute the known values: m( SOT) + 90^ + 90^ + 40^ = 360^ m( SOT) + 220^ = 360^ Subtract 220^ from both sides: m( SOT) = 360^ - 220^ m( SOT) = 140^ Step 3: Find the measure of the inscribed angle TUS. The angle TUS is an inscribed angle that subtends arc ST. The angle SOT is the central angle that subtends the same arc ST. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of the central angle that subtends the same arc. m( TUS) = (1)/(2) m( SOT) Substitute the value of m( SOT): m( TUS) = (1)/(2) × 140^ m( TUS) = 70^ The measure of TUS is 70^.