Given that 𝐵𝑁 equals 11 centimeters, 𝐴𝑀 equals 21 centimeters, and 𝑁𝑀 equals 55 centimeters, find 𝐴𝐵, rounded to the nearest hundredth.
Let’s begin by adding our measurements to the diagram. We are told that 𝐵𝑁 is equal to 11 centimeters, 𝐴𝑀 is equal to 21 centimeters, and 𝑁𝑀 is 55 centimeters. We are asked to find the length of 𝐴𝐵. We can see from our diagram that the line 𝐴𝐵 is a tangent to both circles, which touches the smaller circle at point 𝐵 and the larger circle at point 𝐴. We recall that one of our circle theorems states that a tangent to a circle is perpendicular to the radius at the point of contact. This means that angles 𝑁𝐵𝐴 and 𝑀𝐴𝐵 are both right angles.
In order to calculate the length of 𝐴𝐵, we will firstly draw a parallel line, as shown. This creates a right triangle. Since 𝐵𝑁 is equal to 11 centimeters, 𝐴𝑋 is also 11 centimeters. Since 21 minus 11 is equal to 10, 𝑀𝑋 is equal to 10 centimeters. We can now use the Pythagorean theorem to calculate the length 𝑁𝑋. The Pythagorean theorem states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the two shorter sides of our right triangle.
Substituting in our values, we have 𝑁𝑋 squared plus 10 squared is equal to 55 squared. We know that 10 squared is 100 and 55 squared is 3025. Subtracting 100 from both sides of this equation, we have 𝑁𝑋 squared is equal to 2925. We can then square root both sides. And since 𝑁𝑋 must be positive, this is equal to 15 root 13, which is equal to 54.0832 and so on. As this line is parallel and equal in length to 𝐴𝐵, then 𝐴𝐵 is 54.0832 and so on. We are asked around this to the nearest hundredth. As the third digit after the decimal point is a three, we round down, giving us 54.08.
𝐴𝐵 rounded to the nearest hundredth is 54.08 centimeters.