AI Tutor
Imagine a delivery drone flies along a straight path from a warehouse to a customer's house, but its battery warning beeps when it is exactly 34\frac{3}{4} of the way there. Where on the map grid does it need to land?
To find any spot along a line segment, we do not need to measure the tilted diagonal path directly. We just track how far we travel horizontally and vertically.
What does traveling a fraction of a distance look like in the simplest caseβ€”a flat, one-dimensional number line?

1D Fractional Distance

On a number line from starting point x1x_1 to ending point x2x_2, the total displacement (directed distance) is x2βˆ’x1x_2 - x_1.
To stop at a fraction kk of that journey, take your start and add kk times the total distance: x=x1+k(x2βˆ’x1)x = x_1 + k(x_2 - x_1).
How do we take this simple 1D formula and use it on a 2D coordinate grid with both xx and yy axes?

Moving to the Coordinate Plane

A diagonal path on a grid forms the hypotenuse of a right triangle made of horizontal run (x2βˆ’x1)(x_2 - x_1) and vertical rise (y2βˆ’y1)(y_2 - y_1).
Because similar triangles scale all sides by the exact same factor, traveling a fraction kk along the diagonal means traveling that same fraction kk along the xx-axis and the yy-axis.
This gives us the coordinate formula for a point P(x,y)P(x, y) that is a fraction kk from A(x1,y1)A(x_1, y_1) to B(x2,y2)B(x_2, y_2):