Example 4
Alice, Bob, and Charlie are scuba diving in murky water and would like to determine the visibility (that is, the distance they could be separated before they can no longer see each other). To do this, Alice stays still while Bob and Charlie each swim directly away from her in different directions until they can both see Alice but cannot see each other. Initially, Bob and Charlie are both \(2\) metres from Alice and \(3\) metres from each other.
If Bob and Charlie signal to Alice that they can no longer see each other once they are each \(10\) metres from Alice, what is the visibility?
Solution
Alice, Bob, and Charlie initially form a triangle \(ABC\) with sides lengths \(AB = AC = 2\) m and \(BC = 3\) m.
After Bob and Charlie swim away, they form a new triangle \(AB'C'\) with side lengths \(AB' = AC' = 10\) m.
What is the length of \(B'C'\)?
We can make the following observations:
- the ratios of \(AB'\) to \(AB\) and \(AC'\) to \(AC\) are both equal to \(5\) and
- these pairs of sides contain the shared angle \(A\).
Therefore, triangle \(AB'C'\) is similar to \(ABC\) with a scale factor of \(5\) by the side-angle-side (SAS) similarity rule.
Since triangles are similar, we also know \(B'C'\) should be equal to \(BC\) multiplied by the scale factor
\[ B'C' = 5 \cdot BC = 15 \]
Therefore, the visibility in the water is \(15\) meters.