Boat 1 and boat 2, traveling at constant speeds but not necessarily the same speed, depart at the same time from docks A and C, respectively, on the banks of a man-made circular lake. If they go straight to docks D and B, respectively, they will collide. Prove that if boat 1 goes straight to dock B instead and boat 2 goes straight to dock D, then they arrive at their destinations simultaneously. (Hint: Remember that D = RT.)

Question

Boat 1 and boat 2, traveling at constant speeds but not necessarily the same speed, depart at the same time from docks A and C, respectively, on the banks of a man-made circular lake. If they go straight to docks D and B, respectively, they will collide. Prove that if boat 1 goes straight to dock B instead and boat 2 goes straight to dock D, then they arrive at their destinations simultaneously. (Hint: Remember that D = RT.)

Picture of problem and work I have done so far: gyazo(DOT)com/554696b4f207f8cee425ce84d3200116
I got the triangles similar and am now absolutely lost on what to do.

Answer 1

To prove that boat 1 and boat 2 will arrive at their destinations simultaneously if boat 1 goes to dock B and boat 2 goes to dock D, we can consider the relationships between the distances and speeds.

Let's assume that boat 1 has a constant speed of v1 and boat 2 has a constant speed of v2. Also, let's assume that the distance between docks A and C is d.

Using the formula D = RT (Distance equals Rate times Time), we can determine the time it takes for each boat to reach its destination.

For boat 1:
Time1 = Distance1 / Speed1 = (d + DB) / v1, where DB is the distance from dock B to dock D.

For boat 2:
Time2 = Distance2 / Speed2 = (d + AD) / v2, where AD is the distance from dock A to dock D.

Now, let's consider the triangles formed by the paths of the boats.

Triangle ADC represents the path of boat 2 going from dock A to dock D.

Triangle BCD represents the path of boat 1 going from dock B to dock D.

Since triangle ADC is similar to triangle BCD (with corresponding angles), we can establish the following relationship:

(d + AD) / (d + DB) = v2 / v1

This equation implies that the ratios of the distances are equal to the ratios of the speeds. Therefore, if the speed ratios are equal (v1 / v2 = (d + AD) / (d + DB)), the distances will cancel out, and boat 1 and boat 2 will arrive at their destinations simultaneously.

Hence, if the boats have the same speed ratio (which is equivalent to the ratio of the distances between the docks being the same), boat 1 and boat 2 will reach their destinations at the same time.

Note: It is important to avoid dividing by zero or having a speed equal to zero, as these special cases would require additional considerations and don't apply to the scenario mentioned in the problem.