Answer :
To determine which condition must be true for the parallelogram [tex]\(ABCD\)[/tex] to be a rectangle, we need to verify the specific properties of a rectangle within the context of the coordinates provided. For a shape to be a rectangle, its opposite sides must be equal and parallel, and its adjacent sides must be perpendicular.
Here we analyze the provided options and see which one satisfies the necessary geometric conditions.
### Option A:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_2}=\frac{n_1-n_2}{x_3-x_2}\right) \text{ and } \left(\frac{y_1-\eta_2}{x_4-x_3} \times \frac{n_1-n_2}{x_3-x_2}\right)=-1 \][/tex]
- This equation is checking for some kind of slope relationship, but the indices on the coordinates appear inconsistent with typical format for vertex positions of a rectangle. Moreover, the second part involves a slope product condition that should confirm perpendicular slopes, but the equations don’t properly match the expected format.
### Option B:
[tex]\[ \left(\frac{y_1-n_2}{x_1-x_3}=\frac{n-n_1}{x_2-x_1}\right) \text{ and } \left(\frac{y_4-n}{x_4-x_3} \times \frac{n_2-n_1}{x_1-z_1}\right)=-1 \][/tex]
- Similar to Option A, these equations include an inconsistent use of indices and points. The slope comparisons are irregular and the products don’t accurately confirm perpendicularity for specific sides of the parallelogram.
### Option C:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_2}{x_2-x_1}\right) \text{ and } \left(\frac{y_1-n_1}{x_4-x_3} \times \frac{y_1-n_2}{x_3-x_2}\right)=-1 \][/tex]
- Keeping similar patterns as A and B, these slopes and products do not align with typical criteria to confirm perpendicular relationships in a rectangle.
### Option D:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_1}{x_3-x_1}\right) \text{ and } \left(\frac{y_1-y_1}{x_4-x_3} \times \frac{y_1-n_1}{x_1-x_1}\right)=-1 \][/tex]
- This option directly checks the essential criteria. Let's verify it:
- The first part: [tex]\(\left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_1}{x_3-x_1}\right)\)[/tex] matches respective segments to indicate parallel sides.
- The second part: [tex]\(\left(\frac{y_1-y_1}{x_4-x_3} \times \frac{y_1-n_1}{x_1-x_1}\right)=-1\)[/tex] properly checks for perpendicularity because the product of slopes of adjacent sides in a rectangle should equal [tex]\(-1\)[/tex].
Thus, the expected condition must hold true for [tex]\(ABCD\)[/tex] to be identified as a rectangle under these coordinates and slope verifications, leading us to:
### Correct Answer:
[tex]\[ \boxed{4} \][/tex]
Here we analyze the provided options and see which one satisfies the necessary geometric conditions.
### Option A:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_2}=\frac{n_1-n_2}{x_3-x_2}\right) \text{ and } \left(\frac{y_1-\eta_2}{x_4-x_3} \times \frac{n_1-n_2}{x_3-x_2}\right)=-1 \][/tex]
- This equation is checking for some kind of slope relationship, but the indices on the coordinates appear inconsistent with typical format for vertex positions of a rectangle. Moreover, the second part involves a slope product condition that should confirm perpendicular slopes, but the equations don’t properly match the expected format.
### Option B:
[tex]\[ \left(\frac{y_1-n_2}{x_1-x_3}=\frac{n-n_1}{x_2-x_1}\right) \text{ and } \left(\frac{y_4-n}{x_4-x_3} \times \frac{n_2-n_1}{x_1-z_1}\right)=-1 \][/tex]
- Similar to Option A, these equations include an inconsistent use of indices and points. The slope comparisons are irregular and the products don’t accurately confirm perpendicularity for specific sides of the parallelogram.
### Option C:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_2}{x_2-x_1}\right) \text{ and } \left(\frac{y_1-n_1}{x_4-x_3} \times \frac{y_1-n_2}{x_3-x_2}\right)=-1 \][/tex]
- Keeping similar patterns as A and B, these slopes and products do not align with typical criteria to confirm perpendicular relationships in a rectangle.
### Option D:
[tex]\[ \left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_1}{x_3-x_1}\right) \text{ and } \left(\frac{y_1-y_1}{x_4-x_3} \times \frac{y_1-n_1}{x_1-x_1}\right)=-1 \][/tex]
- This option directly checks the essential criteria. Let's verify it:
- The first part: [tex]\(\left(\frac{y_1-n_1}{x_4-x_3}=\frac{n_1-y_1}{x_3-x_1}\right)\)[/tex] matches respective segments to indicate parallel sides.
- The second part: [tex]\(\left(\frac{y_1-y_1}{x_4-x_3} \times \frac{y_1-n_1}{x_1-x_1}\right)=-1\)[/tex] properly checks for perpendicularity because the product of slopes of adjacent sides in a rectangle should equal [tex]\(-1\)[/tex].
Thus, the expected condition must hold true for [tex]\(ABCD\)[/tex] to be identified as a rectangle under these coordinates and slope verifications, leading us to:
### Correct Answer:
[tex]\[ \boxed{4} \][/tex]
