Answer:
Kim walked a distance of c/2sqrt(2), while Lee walked the same distance.
Explanation:
Let's call the length of the hypotenuse "c", and let's assume that Kim starts at one end of the hypotenuse, while Lee starts at the right angle vertex. We want to find the relationship between the distances that Kim and Lee walked.Since they are walking at the same rate, we can assume that their speeds are equal. Let's call this speed "s".
Let's first find the distance that Kim walked before she reaches the halfway point. Let's call this distance "x". Since the halfway point is exactly halfway along the hypotenuse, we know that the distance between the starting point and the halfway point is also "c/2". Using the Pythagorean theorem, we can write:
x^2 + b^2 = (c/2)^2
where "b" is the distance between the right angle vertex and the halfway point. We can solve for "b" by using the fact that the other leg of the right triangle has length "c/2" (since the halfway point is exactly halfway along the hypotenuse):
b^2 = (c/2)^2 - x^2b = sqrt((c/2)^2 - x^2)
Now, we know that Lee starts from the right angle vertex and walks towards the halfway point. When he reaches the halfway point, the distance he has walked is equal to "b". At this point, he meets Kim, who has walked a distance of "x". Together, they walk the remaining distance from the halfway point to the end of the hypotenuse, which is also "c/2". So, the total distance that Lee walks is:
b + c/2
Substituting for "b", we get:
sqrt((c/2)^2 - x^2) + c/2
Now, we want to find the relationship between the distance that Kim walks (which is "x") and the distance that Lee walks (which we just found). We can substitute the expression we found for "b" into this equation to get:
sqrt((c/2)^2 - x^2) + c/2 = sqrt((c/2)^2 - x^2) + x + c/2 - x
Simplifying, we get:
sqrt((c/2)^2 - x^2) = x
Squaring both sides, we get:
(c/2)^2 - x^2 = x^2
Solving for "x", we get:
x = c/2sqrt(2)
So, Kim walks a distance of "x", which is equal to c/2sqrt(2), and Lee walks a distance of:
sqrt((c/2)^2 - x^2) + c/2 = sqrt((c/2)^2 - (c/2sqrt(2))^2) + c/2 = c/2sqrt(2)
Therefore, the relationship between the distances that Kim and Lee walk is:
Kim walked a distance of c/2sqrt(2), while Lee walked the same distance.