Segment AB is tangent to circle O at B. The diagram is not drawn to scale. If AB = 9 and AO = 12.3, what is the length of the radius (r)? Round your answer to the nearest tenth.

Question

(1 point)
Responses

12.3

8.4

19.6

15.8

GPT-4o · Answer

To find the length of the radius $ r $ of circle O, we need to consider the right triangle formed by the segment $ AB $, the radius $ OB $, and the line segment $ AO $. Here are the given lengths: - $ AB = 9 $ (the tangent to the circle at point B) - $ AO = 12.3 $ (the distance from the center of the circle O to point A) Since segment $ AB $ is tangent to the circle at point B, radius $ OB $ is perpendicular to $ AB $. This gives us a right triangle $ \triangle AOB $ with $ OB $ as one leg (which is the radius $ r $), $ AB $ as the other leg, and $ AO $ as the hypotenuse. We can use the Pythagorean theorem to solve for $ r $: $$ AO^2 = AB^2 + OB^2 $$ $$ (12.3)^2 = (9)^2 + r^2 $$ $$ 151.29 = 81 + r^2 $$ Solving for $ r^2 $: $$ r^2 = 151.29 - 81 $$ $$ r^2 = 70.29 $$ Taking the square root of both sides: $$ r = \sqrt{70.29} \approx 8.4 $$ So, the length of the radius $ r $ is approximately $ 8.4 $, rounded to the nearest tenth.