Suppose line PQ has endpoints P(2,3) and Q(8,-9). Find the coordinates of R and S so that R lies between P and S and line PR is congruent with Line RS which is congruent with line SQ
Let R(x1,y1), S(x2,y2), then
x1=(2*2+8)/3=4, y1=(2*3+(-9))/3=-1
x2=(2+2*8)/3=6, y2=(3+2*(-9))/3=-5
To find the coordinates of points R and S, we'll first need to determine the midpoint of line PQ.
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) can be found using the following formulas:
Midpoint-x: (x₁ + x₂) / 2
Midpoint-y: (y₁ + y₂) / 2
Let's find the midpoint of line PQ:
Midpoint-x = (2 + 8) / 2 = 10 / 2 = 5
Midpoint-y = (3 + (-9)) / 2 = (-6) / 2 = -3
So, the midpoint of line PQ is M(5, -3).
Now that we have the midpoint, R will lie between P and S. To find the coordinates of R, we can use the midpoint formula again:
R's x-coordinate: Midpoint-x = 5
R's y-coordinate: Midpoint-y = -3
Therefore, R(5, -3).
Since line PR is congruent to line RS, we can find the coordinates of S by using the following formula:
S's x-coordinate: R's x-coordinate + (Q's x-coordinate - P's x-coordinate)
S's y-coordinate: R's y-coordinate + (Q's y-coordinate - P's y-coordinate)
Let's calculate S's coordinates:
S's x-coordinate: 5 + (8 - 2) = 5 + 6 = 11
S's y-coordinate: -3 + (-9 - 3) = -3 - 6 = -9
So, S(11, -9).
Therefore, the coordinates of R are (5, -3), and the coordinates of S are (11, -9) to satisfy the given conditions.