In the shown figure, 𝐴𝑍 equals 7.5 centimetres, 𝐵𝑍 equals 14.5 centimetres, and 𝐴𝐶 equals 20 centimetres. Given that all sides of triangle 𝐴𝐵𝐶 are tangent to the shown circle, determine the length of 𝐵𝐶.
Let’s begin by adding the length that we’re given in the question to the diagram. So we have the lengths of two of the sides of this triangle and we’re looking to determine the length of 𝐵𝐶, which is the third side. We’re also given the key piece of information that all three sides of the triangle 𝐴𝐵𝐶 are tangent to the shown circle. Let’s think about what we know about the lengths of tangents drawn from exterior points to circles.
Here’s an important fact. If two segments from the same exterior point are tangent to a circle, then they are congruent. In practice, what this means is that the two line segments drawn from the point 𝐴 to the circle are both equal in length. The same is true for the segments drawn from 𝐵 and those drawn from 𝐶. How would this help us with answering the question? Well, remember we want to determine the length of 𝐵𝐶. So in our diagram, we need to know the length of the pink segment and the length of the green segment.
Using the result we just discussed, we actually already know the length of the pink segment. It’s equal to 𝐵𝑍 which is 14.5 centimetres. So we need to think about how we’re going to calculate the length of the green segment. And to do this, I’m going to add in a couple of labels on the other two sides of this triangle. So just as we have the point 𝑍 on the side 𝐴𝐵, we now have the point 𝑋 on the side 𝐴𝐶 and the point 𝑌 on the side 𝐵𝐶, which are the points where these tangents touch the circle.
We know the full length of the side 𝐴𝐶. But we want to know how much of this is due to the orange part 𝐴𝑋 and how much is due to the green part 𝑋𝐶. Well, applying the same result again, we know that the line segment 𝐴𝑋 is congruent to 𝐴𝑍. And therefore, it’s equal to 7.5 centimetres. The line segment 𝐶𝑋 can therefore be found by subtracting 𝐴𝑋 from the length of 𝐴𝐶: 20 minus 7.5. So we know that 𝐶𝑋 is 12.5 centimetres.
Applying our key result a third time, we know that the two segments drawn from the point 𝐶 are congruent to each other. And therefore, 𝐶𝑌 is congruent to 𝐶𝑋. 𝐶𝑌 is 12.5 centimetres. So our final step in this problem, we need to determine the length of 𝐵𝐶 by summing the two segments 𝐵𝑌 and 𝐶𝑌. So 𝐵𝐶 is equal to 14.5 plus 12.5. It’s 27 centimetres.
The key result which we applied three times within this question is that if two segments from the same exterior point are tangent to a circle, then they are congruent — meaning they’re equal in length.