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Sagot :
To prove that the line segments [tex]\(\overline{AB}\)[/tex] and [tex]\(\overline{CD}\)[/tex] are perpendicular, we need to meet the condition that the product of their slopes is [tex]\(-1\)[/tex].
1. Finding the slopes:
- The slope of [tex]\(\overline{AB}\)[/tex] is given by the change in [tex]\(y\)[/tex] divided by the change in [tex]\(x\)[/tex]:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Similarly, the slope of [tex]\(\overline{CD}\)[/tex] is:
[tex]\[ \text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
2. Condition for perpendicularity:
- For the two line segments to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]:
[tex]\[ \text{slope}_{AB} \times \text{slope}_{CD} = -1 \][/tex]
- Substituting the slopes we found:
[tex]\[ \left( \frac{y_2 - y_1}{x_2 - x_1} \right) \times \left( \frac{y_4 - y_3}{x_4 - x_3} \right) = -1 \][/tex]
Now, we compare this derived condition with the provided options:
- Option A:
[tex]\[ \frac{y_1 - y_2}{x_4 - x_3} \times \frac{y_1 - y_1}{x_3 - x_1} = 1 \][/tex]
This simplifies to [tex]\(0 = 1\)[/tex] (since [tex]\(y_1 - y_1 = 0\)[/tex]), which is incorrect.
- Option B:
[tex]\[ \frac{y_1 - r_2}{y_1 - x_1} + \frac{x_1 - x_1}{x_3 - x_1} = 0 \][/tex]
Since [tex]\(x_1 - x_1 = 0\)[/tex], this simplifies to [tex]\(\frac{y_1 - r_2}{y_1 - x_1} + 0 = 0\)[/tex], which does not represent the required condition of perpendicularity.
- Option C:
[tex]\[ \frac{y_1 - y_1}{x_4 - x_1} \times \frac{y_1 - y_1}{x_3 - x_1} = -1 \][/tex]
However, since [tex]\(y_1 - y_1 = 0\)[/tex], this equates to [tex]\(0 \times 0 = -1\)[/tex], which is not valid.
- Option D:
[tex]\[ \frac{y_2 - y_1}{x_4 - x_3} - \frac{x_2 - x_1}{y_4 - y_3} = 1 \][/tex]
This suggests a difference rather than a product, and is therefore incorrect for checking perpendicularity.
- Option E:
[tex]\[ \frac{y_1 - y_2}{y_3 - x_1} + \frac{x_1 - x_2}{x_3 - x_1} = 0 \][/tex]
This also uses an addition operation rather than the required product.
From the analysis, the correct choice is:
- Option C:
[tex]\[ \frac{(y_2 - y_1)}{(x_2 - x_1)} \times \frac{(y_4 - y_3)}{(x_4 - x_3)} = -1 \][/tex]
This valid formula for perpendicularity matches the derived condition, hence:
The correct answer is Option C.
1. Finding the slopes:
- The slope of [tex]\(\overline{AB}\)[/tex] is given by the change in [tex]\(y\)[/tex] divided by the change in [tex]\(x\)[/tex]:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Similarly, the slope of [tex]\(\overline{CD}\)[/tex] is:
[tex]\[ \text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
2. Condition for perpendicularity:
- For the two line segments to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]:
[tex]\[ \text{slope}_{AB} \times \text{slope}_{CD} = -1 \][/tex]
- Substituting the slopes we found:
[tex]\[ \left( \frac{y_2 - y_1}{x_2 - x_1} \right) \times \left( \frac{y_4 - y_3}{x_4 - x_3} \right) = -1 \][/tex]
Now, we compare this derived condition with the provided options:
- Option A:
[tex]\[ \frac{y_1 - y_2}{x_4 - x_3} \times \frac{y_1 - y_1}{x_3 - x_1} = 1 \][/tex]
This simplifies to [tex]\(0 = 1\)[/tex] (since [tex]\(y_1 - y_1 = 0\)[/tex]), which is incorrect.
- Option B:
[tex]\[ \frac{y_1 - r_2}{y_1 - x_1} + \frac{x_1 - x_1}{x_3 - x_1} = 0 \][/tex]
Since [tex]\(x_1 - x_1 = 0\)[/tex], this simplifies to [tex]\(\frac{y_1 - r_2}{y_1 - x_1} + 0 = 0\)[/tex], which does not represent the required condition of perpendicularity.
- Option C:
[tex]\[ \frac{y_1 - y_1}{x_4 - x_1} \times \frac{y_1 - y_1}{x_3 - x_1} = -1 \][/tex]
However, since [tex]\(y_1 - y_1 = 0\)[/tex], this equates to [tex]\(0 \times 0 = -1\)[/tex], which is not valid.
- Option D:
[tex]\[ \frac{y_2 - y_1}{x_4 - x_3} - \frac{x_2 - x_1}{y_4 - y_3} = 1 \][/tex]
This suggests a difference rather than a product, and is therefore incorrect for checking perpendicularity.
- Option E:
[tex]\[ \frac{y_1 - y_2}{y_3 - x_1} + \frac{x_1 - x_2}{x_3 - x_1} = 0 \][/tex]
This also uses an addition operation rather than the required product.
From the analysis, the correct choice is:
- Option C:
[tex]\[ \frac{(y_2 - y_1)}{(x_2 - x_1)} \times \frac{(y_4 - y_3)}{(x_4 - x_3)} = -1 \][/tex]
This valid formula for perpendicularity matches the derived condition, hence:
The correct answer is Option C.
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