In the figure, chord 𝐴𝐵 of the larger of the concentric circles is tangent to the smaller circle at 𝑋. Given that the radii of the circles are 26 centimeters and 16.1 centimeters, what is the length of line segment 𝐴𝐵 to the nearest tenth of a centimeter?
In the figure, we are given two concentric circles. This means that they have the same center, in this case 𝑚. We are told that 𝐴𝐵 is a chord of the larger circle. It is also a tangent to the smaller circle at the point 𝑋. We are also given the radii of both circles and are asked to calculate the length of the chord 𝐴𝐵.
We will begin by recalling some of the properties of circles known as the circle theorems. Firstly, we know that a tangent to a circle is perpendicular to the radius at the point of contact. This means that the radius 𝑚𝑥 is perpendicular to the tangent 𝐴𝐵. Next, we know that a line from the center of a circle that is perpendicular to a chord also bisects the chord. This means that 𝐴𝑋 is equal to 𝐵𝑋, and the chord 𝐴𝐵 is equal to two multiplied by either one of these.
By joining the points 𝑀 and 𝐵, we create a right triangle 𝑀𝑋𝐵, where 𝑀𝐵 is the radius of the larger circle equal to 26 centimeters and 𝑀𝑋 is the radius of the smaller circle equal to 16.1 centimeters. We can now calculate the length of 𝐵𝑋 using our knowledge of the Pythagorean theorem. This states that, in any right triangle, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse or longest side and 𝑎 and 𝑏 are the lengths of the shorter sides.
In triangle 𝑀𝑋𝐵, we have 𝐵𝑋 squared plus 16.1 squared is equal to 26 squared. 26 squared is 676, and 16.1 squared is 259.21. We can subtract this from both sides of our equation such that 𝐵𝑋 squared is equal to 416.79. Next, we square root both sides. And since 𝐵𝑋 is a length and must be positive, this is equal to 20.4154 and so on.
As already mentioned, the chord 𝐴𝐵 is twice this length. 𝐴𝐵 is therefore equal to 40.830 and so on. We are asked to give our answer to the nearest tenth of a centimeter. And since the second digit after the decimal point is a three, we round down. The length of the chord 𝐴𝐵 to the nearest tenth of a centimeter is therefore equal to 40.8 centimeters.