two perpendicular lines intersect at (2,3) one of the lines has an x-intercept of 2. what are the equations of both lines?
the line through (2,3) and (2,0) is
x = 2
That is a vertical line. So, the perpendicular line through (2,3) is
y = 3
To find the equations of the two perpendicular lines, we need to determine their slopes and y-intercepts.
Let's start with the line that has an x-intercept of 2. The x-intercept represents the point where the line intersects the x-axis, meaning the y-coordinate of this point is 0. Given that the x-intercept is (2, 0), we can substitute these values into the equation y = mx + b, where m is the slope and b is the y-intercept. We have:
0 = 2m + b (equation 1)
Now, let's consider the line which is perpendicular to the first line and intersects at (2, 3). Perpendicular lines have slopes that are negative reciprocals, so the slope of the second line will be the negative reciprocal of the slope of the first line.
We can find the slope of the first line using the formula (y2 - y1) / (x2 - x1), where (x1, y1) = (2, 0) and (x2, y2) = (2, 3):
m1 = (3 - 0) / (2 - 2) = undefined
Since the slope is undefined (or infinite), this indicates that the first line is vertical.
For the second line, given that it is perpendicular to the first line, the slope will be the negative reciprocal of the slope of the first line:
m2 = -1/m1 = -1/undefined = 0
Now that we have the slope of the second line, let's determine its equation:
y = 0x + b (equation 2)
Since the line passes through the point (2, 3), we can substitute these values into equation 2 to find b:
3 = 0(2) + b
3 = 0 + b
b = 3
Therefore, the equation of the second line is y = 0x + 3, or simply y = 3.
In summary, the first line is vertical and does not have a typical equation in the form y = mx + b. The second line is a horizontal line with the equation y = 3.