ANSWER: 45.9 units
Step-by-step explanation:
Given that BC is tangent to Circle A, we can draw a perpendicular line from the center of Circle A to BC. Let's call this line OD, where O is the center of Circle A and D is the point where OD intersects BC.
We know that OD is perpendicular to BC, so we can use the Pythagorean theorem to find its length:
OD^2 = OB^2 - BD^2
where OB is the radius of Circle A.
We're trying to find OB, so let's rearrange the equation:
OB^2 = OD^2 + BD^2
We can find BD using the Pythagorean theorem as well:
BD^2 = AB^2 - AD^2
where AD is half of BC (since OD is perpendicular to BC, it cuts BC in half). So:
AD = BC/2 = 23
BD^2 = AB^2 - AD^2 = 58^2 - 23^2 = 2915
BD ≈ 54.0 (rounded to the nearest tenth)
Now we can substitute this value into our first equation:
OD^2 = OB^2 - BD^2
OD^2 + BD^2 = OB^2
OB ≈ √(OD^2 + BD^2) = √(46^2 + OD^2)
We need to find the value of OD. We know that OD is perpendicular to BC, so triangle ODB is a right triangle. We also know that BD is approximately 54.0. Using the Pythagorean theorem, we can find OD:
OD^2 = OB^2 - BD^2
OD^2 = (√(46^2 + OD^2))^2 - 54.0^2
OD^2 = 2116 + OD^2 - 2916
OD^2 - OD^2 = 2116 - 2916
-OD^2 = -800
OD^2 ≈ 800
OD ≈ 28.3 (rounded to the nearest tenth)
Now we can find the radius:
OB ≈ √(46^2 + OD^2) ≈ √(46^2 + 28.3^2) = √(2111.89) ≈ 45.9
So the length of the radius to the nearest tenth is approximately 45.9 units.