for emma, rate * time = distance becomes 40 * time = distance.
for brandon, rate * time = distance becomes 75 * time = distance.
emma is traveling north.
brandon is traveling east.
the distance that emma is traveling is represented by 40 * time.
the distance that brandon is traveling is represented by 75 * time.
the end points that each is traveling form the hypotenuse of a right triangle.
the length of the hypotenuse is equal to the square root of (emma's distance squared plus brandon's distance squared).
since emma's distance traveled is represented by 40 * time, and brandon's distance traveled is represented by 75 * time, the length of the hypotenuse becomes equal to the square root of (40 * time)^2 + (75 * time)^2)
the distance that each traveled and the distance them will be different depending on the time traveled.
if you allow the distance between them to be represented by y, and you allow the time traveled to be represented by x, then this can be shown graphically.
that graph is shown below, along with the values for distance between them when time is equal to 5 hours and when time is equal to 10 hours.
those values are the coordinate points of (5,425) and (10,850).
the coordinate point of (x,y) = (5,425) says that the distance between them (the y-coordinate) is 425 kilometers per hour when the time (the x-coordinate) is 5 hours.
similar interpretations go for the coordinate point of (x,y) = (10,850).
here's the graph.
here's the picture of the triangle formed that gives you the geometric relationship that creates the formula for the distance between.
the distance between them is the hypotenuse of the right triangle formed.
the formula used to find the length of the distance between them is the pythagorus formula of a^2 + b^2 = c^2, where a and b are legs of the right triangle and c is the hypotenuse of the right triangle.
a represents the distance that emma traveled, which is equal to 40 * time.
b represents the distance that brandon traveled, which is equal to 75 * time.
c represents the distance between them.