Question

Coordinates of the point 2 /3 thirds of the way from (-5,4) and (7,-5)

Answer 1

The coordinates of the point that is two-thirds of the way from (-5,4) to (7,-5) are calculated using linear interpolation to be (3, -2).

Step-by-step explanation:

The student's question pertains to finding the coordinates of a point that is a specific fraction of the way between two given points in a Cartesian coordinate system. This involves using the concept of weighted averages or linear interpolation to calculate the new point's x and y coordinates. To find the point two-thirds of the way from (-5,4) to (7,-5), we perform the following calculations:

1. Calculate the differences in x and y coordinates: Δx = 7 - (-5) = 12, Δy = -5 - 4 = -9.
2. Multiply these differences by the fraction, which is two-thirds: (2/3) × Δx = (2/3) × 12 = 8; (2/3) × Δy = (2/3) × -9 = -6.
3. Add the results to the original point's coordinates: x = -5 + 8 = 3; y = 4 - 6 = -2.

Thus, the coordinates of the point that is two-thirds of the way from (-5,4) to (7,-5) are (3, -2).