Line segment RS is perpendicular to line segment PQ, and the coordinates are R(4, -5), S(-8, 4), P(0, 6), and Q(-3, y). What is the value of y?
First RS
slope = (4 - -5) / (-8 - 4)
= 9/-12 = -3/4
so it is
y = -(3/4) x + b
no need to find b, all we need is the slope -3/4
so
slope of perpendicular = -1/m = +4/3
so
4/3 = (y-6) /( -3 - 0)
4/3 = (y-6)/-3
-4 = y-6
y = 2
To find the value of y, we need to use the fact that line segment RS is perpendicular to line segment PQ.
Perpendicular lines have slopes that are negative reciprocals of each other. So, we can start by finding the slope of line segment PQ.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Using the coordinates of points P(0, 6) and Q(-3, y), we can write the equation to find the slope of PQ:
slope of PQ = (y - 6) / (-3 - 0)
Since line segment RS is perpendicular to line segment PQ, the negative reciprocal of the slope of PQ will be equal to the slope of RS. So, we can write the equation:
slope of RS = -1 / slope of PQ
To find the equation of line RS, we need to use the slope-intercept form: y = mx + b, where m represents the slope and b represents the y-intercept.
We have the coordinates of point R(4, -5), so we can substitute the values into the equation, and solve for b:
-5 = slope of RS * 4 + b
Now, we can calculate the slope of PQ, and then take its negative reciprocal to find the slope of RS.
Finally, substituting the slope of RS and the point R(4, -5) into the equation, we can solve for b, which will give us the equation of line RS.
From the equation of line RS, we can find the y-coordinate of point Q.