Line 1 passes through the points (4, 6) −− and (12,2). Line 2 passes through the points (6, 2 ) −− and (2,0 ) − . Do the lines intersect?
Directions: Complete the following steps on a separate sheet of paper. Be sure to show all of your work:
just calculate the slope of each line
Line 1 has slope -1/2
Line 2 has slope 1/2
since the slopes are different, the lines are not parallel.
So, they must intersect.
Of course, you could just plot the points, draw the lines, and extend them to see that they intersect.
@oobleck, is their a possible equation?
for each line of course
To determine if two lines intersect, we need to find their equations and see if they have a common point of intersection.
Step 1: Find the slope of each line.
The slope formula is given by:
m = (y2 - y1) / (x2 - x1)
For Line 1:
(x1, y1) = (4, 6)
(x2, y2) = (12, 2)
m1 = (2 - 6) / (12 - 4) = -4 / 8 = -1/2
For Line 2:
(x1, y1) = (6, 2)
(x2, y2) = (2, 0)
m2 = (0 - 2) / (2 - 6) = -2 / -4 = 1/2
Step 2: Find the equation of each line.
The equation of a line can be written in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
For Line 1:
Using point-slope form, we have:
(y - y1) = m1(x - x1)
(y - 6) = (-1/2)(x - 4)
y - 6 = (-1/2)x + 2
y = (-1/2)x + 8
For Line 2:
Using point-slope form, we have:
(y - y1) = m2(x - x1)
(y - 2) = (1/2)(x - 6)
y - 2 = (1/2)x - 3
y = (1/2)x - 1
Step 3: Check if the lines intersect.
For lines to intersect, their equations should have a common solution for x and y. So, we need to solve the system of equations formed by the two lines.
(-1/2)x + 8 = (1/2)x - 1
Let's solve this equation to find the x-coordinate of the point of intersection:
Multiply both sides of the equation by 2 to eliminate the fractions:
-2(-1/2)x + 2(8) = 2(1/2)x - 2(1)
x - 16 = x - 2
Subtract x from both sides:
-16 = -2
Since this is not a true statement, we conclude that there is no common solution for x in the equation. Therefore, the lines do not intersect.