Multiple choice

From an external point T, two tangents TM and TN are drawn to a circle with center O, touching the circle at M and N, respectively. Points R and S are taken on the segments TM and TN, respectively, such that the line RS is also tangent to the same circle. If angle ROS = 70°, then the measure of angle RTS, in degrees, is ______.

  1. 10

  2. 40

  3. 70

  4. 90
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the circle touch TM at M, TN at N, and RS at P. Then RM=RP and SN=SP. The perimeter of triangle TRS is TM + TN. The angle ROS = 70 degrees implies angle MON = 140 degrees. The angle RTS = 180 - 140 = 40 degrees.

AI explanation

The angle between a tangent and a chord equals the inscribed angle on the opposite side, making angle MNS = angle NSR. Since angle ROS is 70 degrees, the isosceles triangle NRS has a vertex angle of 180 - 70 = 110 degrees, leaving base angle NSR as (180 - 110) / 2 = 35 degrees. Therefore, angle MNS is 35 degrees, making the angle between the main tangents TM and TN equal to 70 degrees. In quadrilateral OTNS, the radius OT is perpendicular to tangent TM and OS is perpendicular to RS, so the sum of angle TOS and the 70-degree tangent angle is 180 degrees, yielding angle TOS = 110 degrees. The measure of angle RTS is 40 degrees.

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