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.
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.
(1 point)
Responses
12.3
8.4
19.6
15.8
1 answer