Describe positions on the full coordinate grid 4 - Reasoning

Zach's task involves determining the coordinates of point R on a full coordinate grid, given the positions of points P and Q and a relationship between the distances. Point P is located at coordinates (0,4), and point Q is at (3,0). It is stated that the distance from point P to point Q is exactly one third of the distance from P to R. To solve for the coordinates of point R, one must understand that if the distance between P and Q is one third of the distance between P and R, then R must be three times further away from P than Q is, in both the x and y directions.

Given the coordinates of P and Q, we can calculate the distance in the x-direction as 3 units (from 0 to 3) and in the y-direction as 4 units (from 4 to 0). Multiplying these distances by three gives us the distance from P to R, which would be 9 units in the x-direction and 12 units in the y-direction. Since Q lies on the x-axis, and R must be in line with P and Q, R's y-coordinate should be the same as P's but in the opposite direction, making it -8. Therefore, the coordinates of point R are (9,-8), which means R is 9 units to the right of the origin and 8 units below it.