<DOE contains points D(2,3) O(0,0) and E(5,1). Find the coordinates of a point F so that OF is a side of an angle that is adjacent and supplementary to <DOE
It doesn't really matter what point D is. You are looking for a point F such that EOF is a straight line. The line OD is common to both angles.
So, pick F so that its coordinates are on the opposite side of O from E. That is, F=(-5,-1).
To find the coordinates of point F, we'll first need to determine the equation of line OF. Then, we'll find the angle between lines DOE and OF, and use that angle to find point F.
Step 1: Finding the equation of line OF
To find the equation of line OF, we'll need to find the slope of line OF first. We can do this by using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = coordinates of point O(0,0) and (x2, y2) = coordinates of point F.
Let's assign the coordinates of point F as (x, y).
slope OF = (y - y1) / (x - x1) = (y - 0) / (x - 0) = y / x
Since line OF is supposed to be supplementary to line DOE, the slopes of these two lines should multiply to -1. So, the slope of line DOE is:
slope DOE = (y2 - y1) / (x2 - x1) = (1 - 3) / (5 - 2) = -2 / 3
Now, to be supplementary, the product of the slopes should be -1, so we have:
slope DOE * slope OF = -1
(-2/3) * (y/x) = -1
Simplifying this equation, we get:
(-2/3)(y/x) = -1
-2y = -3x
Step 2: Finding the coordinates of point F
Now that we have the equation -2y = -3x, we need one more equation to determine the coordinates of point F. We can use the fact that line OF is adjacent to line DOE. This means that the angle between these two lines is zero degrees, which means they have the same slope.
Slope OF = slope DOE
y / x = -2/3
Since we have two equations now, -2y = -3x and y / x = -2/3, we can solve them simultaneously to find the values of x and y.
From the equation y / x = -2/3, we can substitute y = (-2/3)x into the equation -2y = -3x:
-2(-2/3)x = -3x
4/3x = -3x
Bringing the terms with x on one side, we get:
4/3x + 3x = 0
4x + 9x = 0
13x = 0
Since x = 0, substituting this value back into the equation y = (-2/3)x, we get:
y = (-2/3)(0)
y = 0
Therefore, the coordinates of point F are (0,0).
So, the coordinates of point F so that OF is a side of an angle that is adjacent and supplementary to <DOE are F(0,0).