Find the value of k so that the line containing the points (−4,1) and (k,−2) is perpendicular to the line containing the points (−9,5) and (−5,0).
slope of given line: -5/4
slope of ┴ line: 4/5
(-2-1)/(k+4) = 4/5
solve for k
To find the value of k for which the line containing the points (-4, 1) and (k, -2) is perpendicular to the line containing the points (-9, 5) and (-5, 0), we'll need to use the slope-intercept form of a line.
The slope-intercept form of a line is given by y = mx + b, where m is the slope of the line and b is the y-intercept.
We'll start by finding the slope of the line containing the points (-9, 5) and (-5, 0). The slope of a line between two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Substituting in the coordinates of the given points (-9, 5) and (-5, 0), we get:
m = (0 - 5) / (-5 - (-9))
= -5 / 4
So, the slope of the line containing the points (-9, 5) and (-5, 0) is -5/4.
Since the line we're looking for is perpendicular to this line, the slope of our line will be the negative reciprocal of the slope of the given line.
Let's represent the slope of our line by m1. Therefore, we have:
m1 = -1 / (m)
= -1 / (-5/4)
To divide by a fraction, we invert the fraction and multiply. So we have:
m1 = -1 * (4/-5)
= 4/5
So, the slope of our line is 4/5.
Now, let's use the slope-intercept form to find the equation of the line containing the points (-4, 1) and (k, -2).
Using the point-slope form of a line, the equation of a line with slope m and passing through point (x1, y1) is given by:
y - y1 = m(x - x1)
Substituting the values of (x1, y1) = (-4, 1) and m = 4/5, we get:
y - 1 = (4/5)(x - (-4))
y - 1 = (4/5)(x + 4)
y - 1 = (4/5)x + (4/5)(4)
y - 1 = (4/5)x + 16/5
To find the value of k, we need to find the x-coordinate of the point where this line intersects the y-axis, which is when y is zero.
Setting y = 0, we have:
0 - 1 = (4/5)x + 16/5
-1 - 16/5 = (4/5)x
-21/5 = (4/5)x
Now, solving for x, we get:
x = (-21/5) / (4/5)
= (-21/5) * (5/4)
= -21/4
Therefore, the value of k such that the line containing the points (-4, 1) and (k, -2) is perpendicular to the line containing the points (-9, 5) and (-5, 0) is k = -21/4.