Consider the following diagram, where BC and BD are tangents to the circle, with centre A. The length of each tangent is 4cm and AB=5cm.

Question

Tangents BC and BD meet the circumference of the circle at C and C respectively. A is the midpoint of the circle with radius AC.

What is the radius of the circle? (Do not write the units.)

Answer 1

To find the radius of the circle, we will use the properties of tangents and the geometry of the circle.

1. Identify key points and distances:

   • Let \( O \) be the center of the circle (which is denoted as \( A \) in the problem).
   • The length of each tangent (BC and BD) is 4 cm.
   • The distance from the center \( O \) to a point where the tangent meets the circle (at point \( C \) and \( D \)) is the radius \( r \).

2. Use the properties of tangents:

   • The distance from the center of the circle \( O \) to the point where the tangent touches the circle is perpendicular to the tangent. Therefore, triangle \( OBC \) (and \( OBD \)) is a right triangle.
   • In triangle \( OBC \):
     • \( OB \) is the hypotenuse (which represents \( OA \)), and \( OC \) is the radius \( r \).
     • The tangent \( BC \) is the other leg of the triangle, measuring 4 cm.

3. Apply the Pythagorean theorem: \[ OB^2 = OC^2 + BC^2 \] Substituting the known lengths: \[ AB^2 = r^2 + 4^2 \] Since \( AB = 5 \) cm, \[ 5^2 = r^2 + 4^2 \] \[ 25 = r^2 + 16 \]

4. Solve for the radius \( r \): \[ r^2 = 25 - 16 \] \[ r^2 = 9 \] \[ r = 3 \]

Thus, the radius of the circle is \( 3 \).