AB is tangent to circle O at B. The diagram is not drawn to scale.
if AB =6 and AO =11.7, what is the length of the radius (r)? round answer to the nearest tenth.
a) 5.7
b)8.9
c)10.0
d)13.1
By drawing a diagram and using the fact that AB is tangent to circle O at B, we see that angle AOB is a right angle. Thus, we can use the Pythagorean theorem to find r:
r^2 = AO^2 - AB^2
r^2 = 11.7^2 - 6^2
r^2 = 133.29 - 36
r^2 = 97.29
r ≈ 9.9
So the answer is closest to option (b) 8.9 (rounded to the nearest tenth).
To find the length of the radius (r), we can use the property of a tangent line that states the radius is perpendicular to the tangent line at the point of tangency.
Since AB is tangent to circle O at B, we can draw a right angle triangle with AB as the hypotenuse and the radius (r) as one of the legs.
The other leg of the triangle is the distance from the center of the circle (O) to the point of tangency (B). Let's call this distance x.
Using the Pythagorean theorem, we have:
AB^2 = AO^2 + OB^2
Substituting the given values, we get:
6^2 = 11.7^2 + x^2
Simplifying:
36 = 136.89 + x^2
Rearranging and subtracting 136.89 from both sides:
x^2 = 36 - 136.89
x^2 = -100.89
Since the square of a real number cannot be negative, this equation has no real solution. Therefore, there may be an error in the given information or the diagram.
Thus, we cannot determine the length of the radius (r) based on the given information.
To find the length of the radius (r) of the circle, we can use the properties of a tangent line and a radius at the point of tangency.
In this case, we are given that AB is tangent to the circle at point B. Since AB is tangent to the circle, it forms a right angle with the radius that passes through the point of tangency (B). Let's call the point where the radius meets the circle A.
From the given information, we know that AB is 6 units long and AO (the radius plus AB) is 11.7 units long.
To find the length of the radius (r), we can subtract the length of AB from AO.
So, r = AO - AB = 11.7 - 6 = 5.7
Therefore, the length of the radius (r) is 5.7 units.
Hence, the correct option is a) 5.7.