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.
A. 18.3 cm
B. 20.1 cm
C. 13.2 cm
D. 25.2 cm

Answer 1

To find the length of \( AB \), we can use the Pythagorean theorem because \( A \) (the center of the circle), \( B \), and \( C \) form a right triangle where \( AC \) is the radius, \( BC \) is the tangent, and \( AB \) is the line from the center to the point of tangency.

Let:
- \( AB \) be the length we want to find,
- \( AC \) (the radius) \( = 20 \) cm,
- \( BC \) (the tangent) \( = 15 \) cm.

According to the Pythagorean theorem:

\[
AB^2 + BC^2 = AC^2
\]

Substituting the known values:

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

Calculating the squares:

\[
AB^2 + 225 = 400
\]

Now, isolate \( AB^2 \):

\[
AB^2 = 400 - 225
\]

\[
AB^2 = 175
\]

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

\[
AB = \sqrt{175}
\]

Calculating \( \sqrt{175} \):

\[
\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7} \approx 5 \times 2.64575 \approx 13.228
\]

Rounding to the nearest tenth, we get:

\[
AB \approx 13.2 \text{ cm}
\]

Thus, the length of \( AB \) is approximately \( 13.2 \) cm.

The answer is \( \text{C. } 13.2 \text{ cm} \).