The problem involves a tangent line to a circle, where segment \( AB \) is tangent to circle \( O \) at point \( B \). This means that the radius \( OB \) is perpendicular to the tangent segment \( AB \) at point \( B \).
We can form a right triangle with \( AO \) as the hypotenuse, \( OB \) as one leg, and \( AB \) as the other leg. Let's denote the radius \( OB \) as \( r \).
Using the Pythagorean theorem for the right triangle \( \triangle AOB \):
\[
AO^2 = AB^2 + OB^2
\]
Plugging in the known values \( AO = 12.3 \) and \( AB = 9 \):
\[
12.3^2 = 9^2 + r^2
\]
Calculate \( 12.3^2 \) and \( 9^2 \):
\[
151.29 = 81 + r^2
\]
Now, solve for \( r^2 \):
\[
r^2 = 151.29 - 81
\]
\[
r^2 = 70.29
\]
Taking the square root of both sides to find \( r \):
\[
r = \sqrt{70.29} \approx 8.4
\]
Thus, the length of the radius \( r \) is approximately \( 8.4 \). So the correct response is:
8.4
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