Let A=(0,0), B=(2,4), C=(17,4) and D=(15,0). Then ABCD is a parallelogram. A line through the point (0,−1) divides the parallelogram into two regions of equal area. The slope of this line can be written as a/b where a and b are positive coprime integers. Find a+b.
Let the line from (0,-1) intersect the x-axis at (2+h,0)
We know that h>0 because the line with slope 1/2 passes through (10,4) and obviously falls far short of dividing the are into two equal parts.
So, we now have ABCD divided into two triangles of width 2 and height 4, two rectangles of width h and height 4, and the inside rectangle of width 13-2h and height 4.
By symmetry, we need to have
4(2+h) + 2h = 13
h = 5/6
So, the line from (0,-1) through (17/6,0) will do the equal division.
That line has slope 17/6.
Sorry - the slope is 6/17, but the answer is the same.
To begin, let's visualize the given parallelogram ABCD and the point (0,-1) on a coordinate plane.
First, let's find the equation of line passing through the point (0,-1). We know that the equation of a line can be written in the form y = mx + c, where m is the slope and c is the y-intercept.
Since the line passes through the point (0,-1), we can substitute these coordinates into the equation to solve for c:
-1 = 0 * m + c
-1 = c
So, the equation of the line passing through (0,-1) is y = mx - 1.
Now, we need to find the slope of this line so that we can write it as a/b, where a and b are positive coprime integers. To find the slope, we need two points on the line.
We already have one point that the line passes through, which is (0,-1). Let's find another point on the line by substituting y = mx - 1 into the equation of the line and solve for x.
Let's choose an arbitrary x-coordinate value, let's say x = 1:
y = m * 1 - 1
y = m - 1
So, another point on the line is (1, m - 1).
Now, we can calculate the slope of the line passing through these two points:
m = Δy / Δx
Δy = change in y = (m - 1) - (-1) = m
Δx = change in x = 1 - 0 = 1
m = m / 1 = m
Therefore, the slope of the line passing through (0,-1) and (1, m - 1) is m.
Since we want to write the slope as a fraction of two positive coprime integers, we can use the fraction m/1.
So, a = m and b = 1.
To find the value of m, we can use the concept of equal areas in parallelograms. Since the line divides the parallelogram into two regions of equal area, the line should pass through the midpoint of the diagonal of the parallelogram.
The midpoint of the diagonal BD can be found by adding the coordinates of the endpoints and dividing by 2:
Midpoint of BD = ((0 + 15)/2, (0 + 4)/2) = (7.5, 2)
So, the line passing through (0,-1) should also pass through the point (7.5, 2).
Substituting these coordinates into the equation of the line y = mx - 1:
2 = m * 7.5 - 1
2 + 1 = m * 7.5
3 = 7.5m
m = 3/7.5 = 2/5
Therefore, the slope of the line passing through (0,-1) is 2/5.
Finally, we need to find a + b:
a = 2
b = 5
a + b = 2 + 5 = 7
So, the value of a + b is 7.