<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

If line segment GB is congruent to line segment GE and AC = 15 cm, find the length of segment DF.

To find the coordinates of point F, we can start by calculating the slope of line OE. The slope can be found using the formula:

slope = (y2 - y1) / (x2 - x1)

Using the coordinates of points O(0,0) and E(5,1), we have:

slope(OE) = (1 - 0) / (5 - 0) = 1/5

Since OF is a side of an angle that is adjacent and supplementary to <DOE, we know that the angle between lines OE and OF is 180 degrees.

The slope of line OF can be found by taking the negative reciprocal of the slope of line OE:

slope(OF) = -1 / slope(OE) = -1 / (1/5) = -5

Now, we need to find the y-intercept of line OF. We can use the point-slope form of a line equation:

y - y1 = m(x - x1)

Using point O(0,0) and the slope of OF, we have:

y - 0 = -5(x - 0)
y = -5x

Therefore, the equation of line OF is y = -5x.

To find the coordinates of point F, we need to find the intersection point of lines OE and OF. To do that, we can set the equations of the lines equal to each other and solve for x and y:

-5x = 1/5 * x + 1
-5x - 1/5 * x = 1
-25x - x/5 = 1 * 5
-25x - x = 5
-26x = 5
x = -5/26

Substituting this value of x back into the equation of line OE, we can solve for y:

y = -5 * (-5/26)
y = 25/26

Therefore, the coordinates of point F are (x, y) = (-5/26, 25/26).