1.Is the line contains the points P1(-1,2)and P2(3,4) perpendicular to the line that contains the points Q1(-1,3)and Q2(-4,1)?
2.a line given two points P1(4,5),P2(-4,3)
figure the slopes of each line.
PI,P2 slope=deltay/deltax=4-2/3+1=1/2
P3,P3 slope=3-1/4+4=-1/4
the slopes are not negative reciprocals, they are not perpendicular.
1. To determine if two lines are perpendicular, we can check if the slopes of the lines are negative reciprocals of each other.
Let's find the slope of the first line that contains points P1(-1,2) and P2(3,4):
Slope (m1) = (y2 - y1) / (x2 - x1)
= (4 - 2) / (3 - (-1))
= 2 / 4
= 1/2
Now, let's find the slope of the second line that contains points Q1(-1,3) and Q2(-4,1):
Slope (m2) = (y2 - y1) / (x2 - x1)
= (1 - 3) / (-4 - (-1))
= -2 / -3
= 2/3
Since the slopes (m1 and m2) are not negative reciprocals (1/2 and 2/3) of each other, the lines are NOT perpendicular to each other.
2. To find the equation of a line given two points, we can use the slope-intercept form of a line (y = mx + b), where m is the slope and b is the y-intercept.
Let's find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
= (3 - 5) / (-4 - 4)
= -2 / -8
= 1/4
Now, let's choose one of the points, say P1(4,5), and substitute it into the slope-intercept form:
y = mx + b
5 = (1/4)(4) + b
5 = 1 + b
b = 5 - 1
b = 4
Now we can write the equation of the line:
y = (1/4)x + 4
So the equation of the line passing through points P1(4,5) and P2(-4,3) is y = (1/4)x + 4.