Question

8.
$overline{ab}$ is tangent to circle o at b. what is the length of the radius $r$? round to the nearest tenth.
(there is a diagram with point a, b, o, where ab = 5, ao = 9.8, and ob is the radius $r$ of circle o, and ab is tangent to the circle at b. the diagram is not drawn to scale.)

Explanation:

Step1: Identify the right triangle

Since \( \overline{AB} \) is tangent to circle \( O \) at \( B \), \( OB \perp AB \) (tangent is perpendicular to radius at point of contact). So triangle \( ABO \) is a right triangle with \( \angle ABO = 90^\circ \), \( AB = 5 \), \( AO = 9.8 \), and \( OB = r \).

Step2: Apply Pythagorean theorem

In right triangle \( ABO \), by Pythagorean theorem: \( AO^2 = AB^2 + OB^2 \)
Substitute the known values: \( 9.8^2 = 5^2 + r^2 \)
Calculate \( 9.8^2 = 96.04 \) and \( 5^2 = 25 \), so \( 96.04 = 25 + r^2 \)
Subtract 25 from both sides: \( r^2 = 96.04 - 25 = 71.04 \)
Take square root of both sides: \( r = \sqrt{71.04} \approx 8.4 \)

Answer:

\( 8.4 \)