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Sagot :
Sure, let's find the equation of the perpendicular bisector of the given line segment step-by-step.
1. Determine the slope of the given line:
The equation of the given line is:
[tex]\[ y = -4x - 4 \][/tex]
The slope [tex]\( m \)[/tex] of this line is [tex]\(-4\)[/tex].
2. Find the negative reciprocal of the slope:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. So, we take the negative reciprocal of [tex]\(-4\)[/tex]:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-4} = \frac{1}{4} \][/tex]
Thus, the slope of the perpendicular bisector is [tex]\( \frac{1}{4} \)[/tex].
3. Use the midpoint to find the y-intercept:
The midpoint of the given line segment is given as [tex]\((-1, -2)\)[/tex].
We will use the point-slope form of the equation to find the y-intercept [tex]\( b \)[/tex]. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) = (-1, -2) \)[/tex] and [tex]\( m = \frac{1}{4} \)[/tex].
Substituting these values in, we get:
[tex]\[ y - (-2) = \frac{1}{4}(x - (-1)) \][/tex]
[tex]\[ y + 2 = \frac{1}{4}(x + 1) \][/tex]
Next, let's solve for [tex]\( y \)[/tex]:
[tex]\[ y + 2 = \frac{1}{4}x + \frac{1}{4} \][/tex]
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{4}x + \frac{1}{4} - 2 \][/tex]
Simplify [tex]\( \frac{1}{4} - 2 \)[/tex]:
[tex]\[ \frac{1}{4} - 2 = \frac{1}{4} - \frac{8}{4} = -\frac{7}{4} \][/tex]
So the equation of the perpendicular bisector in slope-intercept form is:
[tex]\[ y = \frac{1}{4}x - \frac{7}{4} \][/tex]
4. Compare with the given choices:
Now we look at the given choices:
[tex]\[ \text{(a) } y = -4x - 4 \][/tex]
[tex]\[ \text{(b) } y = -4x - 6 \][/tex]
[tex]\[ \text{(c) } y = \frac{1}{4}x - 4 \][/tex]
[tex]\[ \text{(d) } y = \frac{1}{4}x - 6 \][/tex]
The correct equation we derived is [tex]\( \frac{1}{4}x - \frac{7}{4} \)[/tex], and it aligns with none of the choices (we would expect choice similar to (c) or (d), but instead between the given options there is no match). As a result:
None of the given choices match the correct perpendicular bisector equation.
1. Determine the slope of the given line:
The equation of the given line is:
[tex]\[ y = -4x - 4 \][/tex]
The slope [tex]\( m \)[/tex] of this line is [tex]\(-4\)[/tex].
2. Find the negative reciprocal of the slope:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. So, we take the negative reciprocal of [tex]\(-4\)[/tex]:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{-4} = \frac{1}{4} \][/tex]
Thus, the slope of the perpendicular bisector is [tex]\( \frac{1}{4} \)[/tex].
3. Use the midpoint to find the y-intercept:
The midpoint of the given line segment is given as [tex]\((-1, -2)\)[/tex].
We will use the point-slope form of the equation to find the y-intercept [tex]\( b \)[/tex]. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, [tex]\( (x_1, y_1) = (-1, -2) \)[/tex] and [tex]\( m = \frac{1}{4} \)[/tex].
Substituting these values in, we get:
[tex]\[ y - (-2) = \frac{1}{4}(x - (-1)) \][/tex]
[tex]\[ y + 2 = \frac{1}{4}(x + 1) \][/tex]
Next, let's solve for [tex]\( y \)[/tex]:
[tex]\[ y + 2 = \frac{1}{4}x + \frac{1}{4} \][/tex]
Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{4}x + \frac{1}{4} - 2 \][/tex]
Simplify [tex]\( \frac{1}{4} - 2 \)[/tex]:
[tex]\[ \frac{1}{4} - 2 = \frac{1}{4} - \frac{8}{4} = -\frac{7}{4} \][/tex]
So the equation of the perpendicular bisector in slope-intercept form is:
[tex]\[ y = \frac{1}{4}x - \frac{7}{4} \][/tex]
4. Compare with the given choices:
Now we look at the given choices:
[tex]\[ \text{(a) } y = -4x - 4 \][/tex]
[tex]\[ \text{(b) } y = -4x - 6 \][/tex]
[tex]\[ \text{(c) } y = \frac{1}{4}x - 4 \][/tex]
[tex]\[ \text{(d) } y = \frac{1}{4}x - 6 \][/tex]
The correct equation we derived is [tex]\( \frac{1}{4}x - \frac{7}{4} \)[/tex], and it aligns with none of the choices (we would expect choice similar to (c) or (d), but instead between the given options there is no match). As a result:
None of the given choices match the correct perpendicular bisector equation.
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