Video Transcript
Circles 𝑀 and 𝑁 have radii of 29 centimeters and 18 centimeters and touch internally at 𝐴. Given that line 𝐴𝐵 is a common tangent to the circles and that triangle 𝐵𝑀𝑁 has an area of 55 square centimeters, what is the length of line segment 𝐴𝐵?
In this question, we are told that line 𝐴𝐵 is a common tangent to the two circles. 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. In this question, this occurs at point 𝐴 as shown. We are told in the question that the radii of the two circles are 29 and 18 centimeters. This means that 𝐴𝑁 equals 18 centimeters and 𝐴𝑀 equals 29 centimeters. We are also told that the area of triangle 𝐵𝑀𝑁 is 55 square centimeters. We can calculate the area of any triangle by multiplying the length of its base by the perpendicular height and then dividing by two.
This means that the area of triangle 𝐵𝑀𝑁 is equal to the base 𝑁𝑀 multiplied by the perpendicular height 𝐴𝐵 divided by two. It is the length of this perpendicular height 𝐴𝐵 that we need to calculate. Since 𝐴𝑀 is equal to 29 centimeters and 𝐴𝑁 is 18 centimeters, we can calculate the length of 𝑁𝑀 by subtracting 18 from 29. This is equal to 11 centimeters. We can now substitute the length of the base and the area of the triangle into our equation. 55 is equal to 11 multiplied by 𝐴𝐵 divided by two. We can multiply both sides of this equation by two. This means that 11 multiplied by 𝐴𝐵 is equal to 110. And dividing through by 11, we have 𝐴𝐵 equals 10. The length of the line segment 𝐴𝐵 is 10 centimeters.
