Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)
To determine if the two lines intersect, we can find the slope of each line and see if they are different.
The slope of the line through (3, 7) and (-2, -5) can be found using the formula:
m = (y2 - y1)/(x2 - x1)
m = (-5 - 7)/(-2 - 3) = (-12)/(-5) = 12/5
The slope of the line through (4, 8) and (10, -2) can be found similarly:
m = (-2 - 8)/(10 - 4) = (-10)/(6) = -5/3
Since the slopes of the two lines are different (12/5 ≠ -5/3), they intersect.
To determine if the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2), we can use the slope-intercept form of a line.
1. Find the slope (m1) of the first line:
- The slope of a line passing through points (x1, y1) and (x2, y2) is given by:
m1 = (y2 - y1) / (x2 - x1)
- Plugging in (3, 7) and (-2, -5), we get:
m1 = (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
2. Find the slope (m2) of the second line:
- Using the same formula, plugging in (4, 8) and (10, -2), we get:
m2 = (-2 - 8) / (10 - 4)
= (-10) / 6
= -5/3
3. If two lines intersect, their slopes will be different.
- If m1 ≠ m2, the lines intersect.
- If m1 = m2, the lines are parallel and do not intersect.
4. Comparing the slopes, m1 = 12/5 and m2 = -5/3, we can see that m1 ≠ m2.
- Therefore, the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2).
To determine if the line through (3, 7) and (-2, -5) intersects with the line through (4, 8) and (10, -2), we can use the equation of a line.
The equation of a line can be written in the form y = mx + b, where m represents the slope of the line and b represents the y-intercept.
First, we need to find the slope (m) of each line. The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
For the first line passing through (3, 7) and (-2, -5):
m1 = (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
For the second line passing through (4, 8) and (10, -2):
m2 = (-2 - 8) / (10 - 4)
= (-10) / 6
= -5/3
Now that we have the slopes for both lines, we can compare them. If the slopes are equal, the lines are parallel and do not intersect. If the slopes are different, the lines will intersect at some point.
In this case, the slopes of the two lines are different (m1 = 12/5 and m2 = -5/3), so we can conclude that the lines do intersect.
Note: Keep in mind that if the lines are parallel and do not intersect, you would need to check if their y-intercepts are also the same. In this case, since we only need to determine if the lines intersect, we do not need to find the y-intercepts.