Answer :
Let's determine if the lines [tex]\( l_1 \)[/tex] and [tex]\( l_2 \)[/tex] given by the points in each question are parallel, perpendicular, or neither.
### Problem 19:
Line [tex]\( l_1 \)[/tex]: Passes through the points [tex]\((-5, -4)\)[/tex] and [tex]\( (0, -5)\)[/tex].
1. Calculate the slope of [tex]\( l_1 \)[/tex]:
Use the slope formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\( l_1 \)[/tex], using the points [tex]\((-5, -4)\)[/tex] and [tex]\((0, -5)\)[/tex], we have:
[tex]\[
\text{slope of } l_1 = \frac{-5 - (-4)}{0 - (-5)} = \frac{-5 + 4}{0 + 5} = \frac{-1}{5} = -0.2
\][/tex]
Line [tex]\( l_2 \)[/tex]: Passes through the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 1)\)[/tex].
2. Calculate the slope of [tex]\( l_2 \)[/tex]:
For [tex]\( l_2 \)[/tex], using the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 1)\)[/tex], we have:
[tex]\[
\text{slope of } l_2 = \frac{1 - (-2)}{2 - 1} = \frac{1 + 2}{2 - 1} = \frac{3}{1} = 3
\][/tex]
Comparison:
3. Are the lines parallel?
Lines are parallel if their slopes are equal. Here, the slopes are [tex]\(-0.2\)[/tex] and [tex]\(3\)[/tex]. Since these are not equal, [tex]\( l_1 \)[/tex] and [tex]\( l_2 \)[/tex] are not parallel.
4. Are the lines perpendicular?
Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Let's check:
[tex]\[
(-0.2) \times 3 = -0.6
\][/tex]
Since [tex]\(-0.6\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
Conclusion for Problem 19:
The lines are neither parallel nor perpendicular.
### Problem 20:
Line [tex]\( l_2 \)[/tex]: Considering its points are [tex]\((-2, -1)\)[/tex] and [tex]\( (0, -4)\)[/tex].
1. Calculate the slope of [tex]\( l_2 \)[/tex]:
[tex]\[
\text{slope of } l_2 = \frac{-4 - (-1)}{0 - (-2)} = \frac{-4 + 1}{0 + 2} = \frac{-3}{2} = -1.5
\][/tex]
Line [tex]\( l_2 \)[/tex] [again]: Considering points [tex]\((-3, 3)\)[/tex] and [tex]\( (3, 1)\)[/tex].
2. Calculate the slope of the other [tex]\( l_2 \)[/tex]:
[tex]\[
\text{slope of } l_2 = \frac{1 - 3}{3 - (-3)} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
Comparison:
3. Are the lines parallel?
Check the slopes: [tex]\(-1.5\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex]. Since the slopes are not equal, the lines are not parallel.
4. Are the lines perpendicular?
Calculate the product of the slopes:
[tex]\[
(-1.5) \times \left(-\frac{1}{3}\right) = 0.5
\][/tex]
Since [tex]\(0.5\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
Conclusion for Problem 20:
The lines are neither parallel nor perpendicular.
### Problem 19:
Line [tex]\( l_1 \)[/tex]: Passes through the points [tex]\((-5, -4)\)[/tex] and [tex]\( (0, -5)\)[/tex].
1. Calculate the slope of [tex]\( l_1 \)[/tex]:
Use the slope formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For [tex]\( l_1 \)[/tex], using the points [tex]\((-5, -4)\)[/tex] and [tex]\((0, -5)\)[/tex], we have:
[tex]\[
\text{slope of } l_1 = \frac{-5 - (-4)}{0 - (-5)} = \frac{-5 + 4}{0 + 5} = \frac{-1}{5} = -0.2
\][/tex]
Line [tex]\( l_2 \)[/tex]: Passes through the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 1)\)[/tex].
2. Calculate the slope of [tex]\( l_2 \)[/tex]:
For [tex]\( l_2 \)[/tex], using the points [tex]\((1, -2)\)[/tex] and [tex]\((2, 1)\)[/tex], we have:
[tex]\[
\text{slope of } l_2 = \frac{1 - (-2)}{2 - 1} = \frac{1 + 2}{2 - 1} = \frac{3}{1} = 3
\][/tex]
Comparison:
3. Are the lines parallel?
Lines are parallel if their slopes are equal. Here, the slopes are [tex]\(-0.2\)[/tex] and [tex]\(3\)[/tex]. Since these are not equal, [tex]\( l_1 \)[/tex] and [tex]\( l_2 \)[/tex] are not parallel.
4. Are the lines perpendicular?
Lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex]. Let's check:
[tex]\[
(-0.2) \times 3 = -0.6
\][/tex]
Since [tex]\(-0.6\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
Conclusion for Problem 19:
The lines are neither parallel nor perpendicular.
### Problem 20:
Line [tex]\( l_2 \)[/tex]: Considering its points are [tex]\((-2, -1)\)[/tex] and [tex]\( (0, -4)\)[/tex].
1. Calculate the slope of [tex]\( l_2 \)[/tex]:
[tex]\[
\text{slope of } l_2 = \frac{-4 - (-1)}{0 - (-2)} = \frac{-4 + 1}{0 + 2} = \frac{-3}{2} = -1.5
\][/tex]
Line [tex]\( l_2 \)[/tex] [again]: Considering points [tex]\((-3, 3)\)[/tex] and [tex]\( (3, 1)\)[/tex].
2. Calculate the slope of the other [tex]\( l_2 \)[/tex]:
[tex]\[
\text{slope of } l_2 = \frac{1 - 3}{3 - (-3)} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
Comparison:
3. Are the lines parallel?
Check the slopes: [tex]\(-1.5\)[/tex] and [tex]\(-\frac{1}{3}\)[/tex]. Since the slopes are not equal, the lines are not parallel.
4. Are the lines perpendicular?
Calculate the product of the slopes:
[tex]\[
(-1.5) \times \left(-\frac{1}{3}\right) = 0.5
\][/tex]
Since [tex]\(0.5\)[/tex] is not equal to [tex]\(-1\)[/tex], the lines are not perpendicular.
Conclusion for Problem 20:
The lines are neither parallel nor perpendicular.
