Answer :
To determine which of the given statements is true for ensuring that a parallelogram [tex]\(ABCD\)[/tex] is a rectangle, let's break down the necessary conditions for a parallelogram to be a rectangle:
1. Opposite Sides are Parallel:
- This is already true by the definition of a parallelogram.
2. Each Angle is 90 Degrees:
- This requires that adjacent sides be perpendicular to each other. In mathematical terms, if the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
For a rectangle, we need to ensure both the parallelogram and perpendicularity conditions. Here, we focus on ensuring perpendicularity.
Let's use the points provided:
- [tex]\( A(x_1, y_1) \)[/tex]
- [tex]\( B(x_2, y_2) \)[/tex]
- [tex]\( C(x_3, y_3) \)[/tex]
- [tex]\( D(x_4, y_4) \)[/tex]
### Step-by-Step Analysis
#### Calculate Slopes
Slope between two points [tex]\((x_i, y_i)\)[/tex] and [tex]\((x_j, y_j)\)[/tex] is given by:
[tex]\[ \text{slope} (i,j) = \frac{y_j - y_i}{x_j - x_i} \][/tex]
Here's what we calculate:
1. Slope [tex]\(AB\)[/tex]:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
2. Slope [tex]\(BC\)[/tex]:
[tex]\[ \text{slope}_{BC} = \frac{y_3 - y_2}{x_3 - x_2} \][/tex]
3. Slope [tex]\(CD\)[/tex]:
[tex]\[ \text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
4. Slope [tex]\(DA\)[/tex]:
[tex]\[ \text{slope}_{DA} = \frac{y_1 - y_4}{x_1 - x_4} \][/tex]
#### Ensure Perpendicularity
We need the slopes of adjacent sides to be negative reciprocals:
1. Perpendicular Between AB and BC:
[tex]\[ \text{slope}_{AB} \times \text{slope}_{BC} = -1 \][/tex]
2. Perpendicular Between BC and CD:
[tex]\[ \text{slope}_{BC} \times \text{slope}_{CD} = -1 \][/tex]
#### Check the Options
Option A:
[tex]\[ \left(\frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_2}{x_3 - x_2}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2}\right) = -1 \][/tex]
- The first part ([tex]\(\text{slope}_{CD} = \text{slope}_{BC}\)[/tex]) is incorrect because it contradicts the definition of perpendicular slopes. Parallel slopes won't work for perpendicularity.
- The second part is incorrect, as correct negative reciprocals should be involved.
Option B:
[tex]\[ \left(\frac{y_4 - y_1}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1}\right) = -1 \][/tex]
- The first part is confusing with indices.
- It’s incorrect to test the perpendicularity this way.
Option C:
[tex]\[ \left(\frac{y_4 - y_1}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_1 - y_2}{x_3 - x_2}\right) = -1 \][/tex]
- The first part is logically testing slopes properly for perpendicular tests.
- The second part correctly checks perpendicular slopes.
Option D:
[tex]\[ \left(\frac{y_4 - y_3}{x_4 - x_3} = \frac{y_1 - y_2}{x_3 - x_1}\right) \text{ and } \left( \frac{y_4 - y_1}{x_4 - x_3} \times \frac{y_2 - y_2}{x_2 - x_1}\right) = -1 \][/tex]
- Contains incorrect slopes and unclear expressions that don’t directly contribute to perpendicular conditions.
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
Because it correctly describes ensuring perpendicularity and logical slope checks.
1. Opposite Sides are Parallel:
- This is already true by the definition of a parallelogram.
2. Each Angle is 90 Degrees:
- This requires that adjacent sides be perpendicular to each other. In mathematical terms, if the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
For a rectangle, we need to ensure both the parallelogram and perpendicularity conditions. Here, we focus on ensuring perpendicularity.
Let's use the points provided:
- [tex]\( A(x_1, y_1) \)[/tex]
- [tex]\( B(x_2, y_2) \)[/tex]
- [tex]\( C(x_3, y_3) \)[/tex]
- [tex]\( D(x_4, y_4) \)[/tex]
### Step-by-Step Analysis
#### Calculate Slopes
Slope between two points [tex]\((x_i, y_i)\)[/tex] and [tex]\((x_j, y_j)\)[/tex] is given by:
[tex]\[ \text{slope} (i,j) = \frac{y_j - y_i}{x_j - x_i} \][/tex]
Here's what we calculate:
1. Slope [tex]\(AB\)[/tex]:
[tex]\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
2. Slope [tex]\(BC\)[/tex]:
[tex]\[ \text{slope}_{BC} = \frac{y_3 - y_2}{x_3 - x_2} \][/tex]
3. Slope [tex]\(CD\)[/tex]:
[tex]\[ \text{slope}_{CD} = \frac{y_4 - y_3}{x_4 - x_3} \][/tex]
4. Slope [tex]\(DA\)[/tex]:
[tex]\[ \text{slope}_{DA} = \frac{y_1 - y_4}{x_1 - x_4} \][/tex]
#### Ensure Perpendicularity
We need the slopes of adjacent sides to be negative reciprocals:
1. Perpendicular Between AB and BC:
[tex]\[ \text{slope}_{AB} \times \text{slope}_{BC} = -1 \][/tex]
2. Perpendicular Between BC and CD:
[tex]\[ \text{slope}_{BC} \times \text{slope}_{CD} = -1 \][/tex]
#### Check the Options
Option A:
[tex]\[ \left(\frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_2}{x_3 - x_2}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2}\right) = -1 \][/tex]
- The first part ([tex]\(\text{slope}_{CD} = \text{slope}_{BC}\)[/tex]) is incorrect because it contradicts the definition of perpendicular slopes. Parallel slopes won't work for perpendicularity.
- The second part is incorrect, as correct negative reciprocals should be involved.
Option B:
[tex]\[ \left(\frac{y_4 - y_1}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1}\right) = -1 \][/tex]
- The first part is confusing with indices.
- It’s incorrect to test the perpendicularity this way.
Option C:
[tex]\[ \left(\frac{y_4 - y_1}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1}\right) \text{ and } \left(\frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_1 - y_2}{x_3 - x_2}\right) = -1 \][/tex]
- The first part is logically testing slopes properly for perpendicular tests.
- The second part correctly checks perpendicular slopes.
Option D:
[tex]\[ \left(\frac{y_4 - y_3}{x_4 - x_3} = \frac{y_1 - y_2}{x_3 - x_1}\right) \text{ and } \left( \frac{y_4 - y_1}{x_4 - x_3} \times \frac{y_2 - y_2}{x_2 - x_1}\right) = -1 \][/tex]
- Contains incorrect slopes and unclear expressions that don’t directly contribute to perpendicular conditions.
Thus, the correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
Because it correctly describes ensuring perpendicularity and logical slope checks.
