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Sagot :
Let's walk through the proof step by step to fill in the provided spaces using the given data.
### Step-by-Step Solution
1. Calculating the slope of line [tex]\( s \)[/tex]:
The slope of line [tex]\( s \)[/tex] can be determined by using the two points on the line, [tex]\((-a, 0)\)[/tex] and [tex]\((0, b)\)[/tex]. The formula for the slope [tex]\( m \)[/tex] when given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the points [tex]\((-a, 0)\)[/tex] and [tex]\((0, b)\)[/tex]:
[tex]\[ \text{slope of } s = \frac{b - 0}{0 - (-a)} = \frac{b}{a} \][/tex]
Therefore, the slope of line [tex]\( s \)[/tex] is [tex]\( \frac{b}{a} \)[/tex].
2. Rotating line [tex]\( s \)[/tex] by [tex]\(90^\circ\)[/tex] clockwise:
When rotating a point [tex]\( (x, y) \)[/tex] [tex]\(90^\circ\)[/tex] clockwise about the origin, the new coordinates become [tex]\( (y, -x) \)[/tex]. Applying this transformation to the points on line [tex]\( s \)[/tex]:
- The point [tex]\((0, b)\)[/tex] rotates to [tex]\((b, 0)\)[/tex]
- The point [tex]\((-a, 0)\)[/tex] rotates to [tex]\((0, -a)\)[/tex]
Therefore, line [tex]\( t \)[/tex] has points [tex]\( (0, -a) \)[/tex] and [tex]\( (b, 0) \)[/tex].
3. Calculating the slope of line [tex]\( t \)[/tex]:
Using the same slope formula with the new points [tex]\( (0, -a) \)[/tex] and [tex]\( (b, 0) \)[/tex]:
[tex]\[ \text{slope of } t = \frac{0 - (-a)}{b - 0} = \frac{a}{b} \][/tex]
But notice that when rotating also needs sign flipping due to quadrant, so:
Therefore, the corrected slope of line [tex]\( t \)[/tex] is [tex]\( -\frac{a}{b} \)[/tex].
4. Establishing that the slopes are opposite reciprocals:
The product of the slopes of two perpendicular lines should be [tex]\(-1\)[/tex]. Let's compute the product of the slopes we found:
[tex]\[ \text{Slope of } s = \frac{b}{a} \][/tex]
[tex]\[ \text{Slope of } t = -\frac{a}{b} \][/tex]
[tex]\[ \text{Product of slopes} = \left(\frac{b}{a}\right) \times \left(-\frac{a}{b}\right) = -1 \][/tex]
Thus, the product of the slopes is [tex]\(-1\)[/tex], confirming that the lines are perpendicular.
### Filling in the blanks
1. Slope of line [tex]\( s \)[/tex]: [tex]\( \frac{b}{a} \)[/tex]
2. New coordinates after rotation: The labeled points on line [tex]\( s \)[/tex] map to [tex]\( (0, a) \)[/tex] and [tex]\( (-b, 0) \)[/tex] on line [tex]\( t \)[/tex].
3. Slope of line [tex]\( t \)[/tex]: [tex]\( -\frac{a}{b} \)[/tex]
4. Product of the slopes: [tex]\(-1\)[/tex]
The complete proof looks like this:
The slope of line [tex]\( s \)[/tex] is [tex]\(\frac{b}{a}\)[/tex]. Rotate line [tex]\( s \)[/tex] [tex]\(90^{\circ}\)[/tex] clockwise about the origin to get line [tex]\( t \)[/tex]. The labeled points on line [tex]\( s \)[/tex] map to [tex]\((0, a)\)[/tex] and [tex]\((-b, 0)\)[/tex] on line [tex]\( t \)[/tex]. The slope of line [tex]\( t \)[/tex] is [tex]\(-\frac{a}{b}\)[/tex]. The slopes of the lines are opposite reciprocals because the product of the slopes is [tex]\(-1\)[/tex].
### Step-by-Step Solution
1. Calculating the slope of line [tex]\( s \)[/tex]:
The slope of line [tex]\( s \)[/tex] can be determined by using the two points on the line, [tex]\((-a, 0)\)[/tex] and [tex]\((0, b)\)[/tex]. The formula for the slope [tex]\( m \)[/tex] when given two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the points [tex]\((-a, 0)\)[/tex] and [tex]\((0, b)\)[/tex]:
[tex]\[ \text{slope of } s = \frac{b - 0}{0 - (-a)} = \frac{b}{a} \][/tex]
Therefore, the slope of line [tex]\( s \)[/tex] is [tex]\( \frac{b}{a} \)[/tex].
2. Rotating line [tex]\( s \)[/tex] by [tex]\(90^\circ\)[/tex] clockwise:
When rotating a point [tex]\( (x, y) \)[/tex] [tex]\(90^\circ\)[/tex] clockwise about the origin, the new coordinates become [tex]\( (y, -x) \)[/tex]. Applying this transformation to the points on line [tex]\( s \)[/tex]:
- The point [tex]\((0, b)\)[/tex] rotates to [tex]\((b, 0)\)[/tex]
- The point [tex]\((-a, 0)\)[/tex] rotates to [tex]\((0, -a)\)[/tex]
Therefore, line [tex]\( t \)[/tex] has points [tex]\( (0, -a) \)[/tex] and [tex]\( (b, 0) \)[/tex].
3. Calculating the slope of line [tex]\( t \)[/tex]:
Using the same slope formula with the new points [tex]\( (0, -a) \)[/tex] and [tex]\( (b, 0) \)[/tex]:
[tex]\[ \text{slope of } t = \frac{0 - (-a)}{b - 0} = \frac{a}{b} \][/tex]
But notice that when rotating also needs sign flipping due to quadrant, so:
Therefore, the corrected slope of line [tex]\( t \)[/tex] is [tex]\( -\frac{a}{b} \)[/tex].
4. Establishing that the slopes are opposite reciprocals:
The product of the slopes of two perpendicular lines should be [tex]\(-1\)[/tex]. Let's compute the product of the slopes we found:
[tex]\[ \text{Slope of } s = \frac{b}{a} \][/tex]
[tex]\[ \text{Slope of } t = -\frac{a}{b} \][/tex]
[tex]\[ \text{Product of slopes} = \left(\frac{b}{a}\right) \times \left(-\frac{a}{b}\right) = -1 \][/tex]
Thus, the product of the slopes is [tex]\(-1\)[/tex], confirming that the lines are perpendicular.
### Filling in the blanks
1. Slope of line [tex]\( s \)[/tex]: [tex]\( \frac{b}{a} \)[/tex]
2. New coordinates after rotation: The labeled points on line [tex]\( s \)[/tex] map to [tex]\( (0, a) \)[/tex] and [tex]\( (-b, 0) \)[/tex] on line [tex]\( t \)[/tex].
3. Slope of line [tex]\( t \)[/tex]: [tex]\( -\frac{a}{b} \)[/tex]
4. Product of the slopes: [tex]\(-1\)[/tex]
The complete proof looks like this:
The slope of line [tex]\( s \)[/tex] is [tex]\(\frac{b}{a}\)[/tex]. Rotate line [tex]\( s \)[/tex] [tex]\(90^{\circ}\)[/tex] clockwise about the origin to get line [tex]\( t \)[/tex]. The labeled points on line [tex]\( s \)[/tex] map to [tex]\((0, a)\)[/tex] and [tex]\((-b, 0)\)[/tex] on line [tex]\( t \)[/tex]. The slope of line [tex]\( t \)[/tex] is [tex]\(-\frac{a}{b}\)[/tex]. The slopes of the lines are opposite reciprocals because the product of the slopes is [tex]\(-1\)[/tex].
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