Given the points A (1,8), B(4,7) and O(0,0). Find the coordinates of the points C at which these two lines intersect?

To find the coordinates of point C where the lines AB and OC intersect, we first need to find the equations of the lines AB and OC.

The equation of the line passing through points A(1,8) and B(4,7) can be found using the point-slope formula:

m = (y2 - y1) / (x2 - x1)
m = (7 - 8) / (4 - 1)
m = -1 / 3

Now, using the point-slope formula with point A(1,8):

y - 8 = -1/3(x - 1)
y - 8 = -1/3x + 1/3
y = -1/3x + 25/3

So, the equation of line AB is y = -1/3x + 25/3.

Now, let's find the equation of the line passing through points O(0,0) and C(x, y). Since O is the origin, the equation of line OC is simply y = mx, where m is the slope of the line.

The slope of line OC can be found using the coordinates of points O and C:

m = (y - 0) / (x - 0)
m = y / x

Now, we can set the equations of line AB and OC equal to each other to find the coordinates of point C:

-1/3x + 25/3 = y / x

Multiplying both sides by 3x to get rid of the fractions:

- x^2 + 25 = 3y

Rearranging the equation:

x^2 + 3y - 25 = 0

Now, we can substitute the coordinates of point A(1,8) into the equation to find the coordinates of point C:

1 + 3(8) - 25 = 0
1 + 24 - 25 = 0
0 = 0

Therefore, the coordinates of point C are (1, 8).

The equation of the line passing through A(1,8) and B(4,7) is:

(y - 8) / (x - 1) = (7 - 8) / (4 - 1)
y - 8 = -1/3 - (x - 1)
y = -x/3 + 5

The equation of the line passing through O(0,0) and C(x,y) is:

(y - 0) / (x - 0) = (y - 7) / (x - 4)
y/x = (y - 7) / (x - 4)
xy - 4y = x(y - 7)
xy - 4y = xy - 7x
7x = 4y
x/y = 4/7

Substituting x/y = 4/7 in the equation of the first line, we get:

y = -4/7 - x + 5

Solving for x when y = 0, we get:

0 = -4/7 - x + 5
x = 5 - 7/4 = 35/4

Substituting x = 35/4 in the equation of the line passing through O and C, we get:

y = (35/4) - 4/7 = 5

Therefore, the coordinates of the point C are (35/4, 5).