Video Transcript
In the circles with centers 𝑀 and 𝑁, which are tangent to one another at 𝐴 shown below, line segment 𝑀𝐵 is a tangent to the circle 𝑁. 𝑀𝐶 is equal to 24 centimeters, and 𝑀𝑁 is equal to 25 centimeters. Find the length of the line segment 𝐶𝐵.
In this question, we are told that the two circles are tangent to one another at 𝐴. This means that they touch internally at point 𝐴. We are told that 𝑀𝐵 is a tangent to circle 𝑁 and touches the circle at point 𝐶. Length 𝑀𝐶 is equal to 24 centimeters, and length 𝑀𝑁 is equal to 25 centimeters. We are asked to find the length of the line segment 𝐶𝐵.
We can see from the figure that 𝑀𝐵 is the radius of the larger circle. This means that 𝑀𝐵 is equal to 𝑀𝐶 plus 𝐶𝐵. As already mentioned, 𝑀𝐶 is equal to 24 centimeters. This means that if we can calculate the length of the radius of the larger circle, we can use it to work out the value of 𝐶𝐵.
We notice from the figure that triangle 𝑁𝐶𝑀 is a right triangle. We know this as a tangent to a circle is perpendicular to the radius at the point of contact. 𝑁𝐶 is the radius of the smaller circle. And we can calculate this length 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 longest side, known as the hypotenuse, and 𝐴 and 𝐵 are the lengths of the shorter sides.
Substituting in our values, we have 𝑁𝐶 squared plus 24 squared is equal to 25 squared. 24 squared is 576, and 25 squared is 625. Subtracting 576 from both sides of this equation gives us 𝑁𝐶 squared is equal to 49. We can then square root both sides. And since 𝑁𝐶 is a length, it must be positive. The radius of the smaller circle 𝑁𝐶 is equal to seven centimeters.
It is worth noting that this is one of our Pythagorean triples. Any triangle with side lengths in the ratio seven-24-25 will be a right triangle. Clearing some space, we note that if 𝑁𝐶 equals seven centimeters, then 𝑁𝐴 must also be seven centimeters, as the radius of a circle is the distance from the center to any point on its circumference. The length 𝑀𝐴 must therefore be equal to 25 centimeters plus seven centimeters. This is equal to 32 centimeters and is the radius of the larger circle. We know that 𝑀𝐵 is also a radius of the larger circle. Therefore, it too must be equal to 32 centimeters.
We now have values of 𝑀𝐵 and 𝑀𝐶, which we can use to calculate the length of 𝐶𝐵. We can solve the equation 32 is equal to 24 plus 𝐶𝐵. We do this by subtracting 24 from both sides such that 𝐶𝐵 is equal to eight. The length of the line segment 𝐶𝐵 is eight centimeters.
