We are given a circle with center O and a tangent line AB to the circle at point B. The length of AB is 9, and the length of AO is 10.5. We want to find the length of the radius r of the circle.

1. Problem Description

We are given a circle with center O and a tangent line AB to the circle at point B. The length of AB is 9, and the length of AO is 10.

5. We want to find the length of the radius r of the circle.

2. Solution Steps

Since AB is tangent to the circle at B, the radius OB is perpendicular to AB. Therefore, triangle ABO is a right triangle with a right angle at B.

We can use the Pythagorean theorem to find the length of OB, which is the radius r of the circle.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In triangle ABO, AO is the hypotenuse, and AB and OB are the other two sides. So, we have:

$AO^2 = AB^2 + OB^2$

We are given that $AO = 10.5$ and $AB = 9$. We want to find $OB = r$.

Substituting the given values into the Pythagorean theorem, we have:

$(10.5)^2 = (9)^2 + r^2$

$110.25 = 81 + r^2$

$r^2 = 110.25 - 81$

$r^2 = 29.25$

$r = \sqrt{29.25}$

$r \approx 5.4083269$

Rounding to the nearest tenth, we get $r \approx 5.4$.

3. Final Answer

5. 4