Question

Point A is the center of a circle and BC is a tangent to the circle. BC = 15 cm and AC = 20 cm. What is the length of AB? Round to the nearest tenth.

Answer 1

To find the length of segment AB, we can apply the Pythagorean theorem. Here, we have the following information:

- A is the center of the circle.
- BC is a tangent to the circle, and the length of BC is 15 cm.
- AC is the radius from the center A to point C on the circle, with a length of 20 cm.

Since BC is a tangent to the circle, we can conclude that \(AB\) (the line from the center of the circle to the point of tangency B) is perpendicular to the tangent line BC at point B. Therefore, triangle ABC forms a right triangle with:

- AC as one leg (20 cm),
- BC as the other leg (15 cm), and
- AB as the hypotenuse.

According to the Pythagorean theorem:
\[
AC^2 = AB^2 + BC^2
\]
Substituting the known lengths:

\[
20^2 = AB^2 + 15^2
\]

Calculating the squares:

\[
400 = AB^2 + 225
\]

Now, subtract 225 from both sides:

\[
400 - 225 = AB^2
\]
\[
175 = AB^2
\]

To find \(AB\), take the square root of both sides:

\[
AB = \sqrt{175}
\]

Calculating the square root gives:

\[
AB \approx 13.3 \text{ cm (rounding to the nearest tenth)}
\]

Therefore, the length of AB is approximately \( \boxed{13.3} \) cm.