Answer :
To determine which of the conditions must be true for the parallelogram [tex]\(ABCD\)[/tex] to be a rectangle, we need to consider the properties of both rectangles and parallelograms.
1. Properties of a Rectangle:
- A rectangle is a type of parallelogram where each angle is [tex]\(90^\circ\)[/tex].
- For each internal angle to be [tex]\(90^\circ\)[/tex], any two adjacent sides must be perpendicular to each other.
2. Properties of Slopes:
- The slope of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
- Two lines are perpendicular if and only if the product of their slopes is [tex]\(-1\)[/tex].
Given these properties, let’s analyze the slopes of the given sides in the parallelogram [tex]\(ABCD\)[/tex]:
For parallelogram [tex]\(ABCD\)[/tex] to be a rectangle:
- The slopes of opposite sides must be equal.
- Adjacent sides must be perpendicular to each other.
We will denote the following slopes:
- Slope of [tex]\(AD\)[/tex]: [tex]\(\frac{y_4 - y_3}{x_4 - x_3}\)[/tex]
- Slope of [tex]\(AB\)[/tex]: [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]
Then, for [tex]\(ABCD\)[/tex] to be a rectangle, two conditions must hold:
1. Opposite sides must have equal slopes. Hence, [tex]\( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex].
2. Adjacent sides must be perpendicular, which means [tex]\( \left( \frac{y_4 - y_3}{x_4 - x_3} \right) \times \left( \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \)[/tex].
Now, let’s look at each of the provided options:
- Option A:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_2}{x_3 - x_2} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
- This option contains an incorrect duplicate [tex]\( \frac{y_2 - y_2}{x_3 - x_2} \)[/tex], which simplifies incorrectly.
- Option B:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \][/tex]
- This option exactly matches our conditions: equal slopes of opposite sides and perpendicular adjacent sides.
- Option C:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_{1}}{x_2 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
- The second condition involves a different slope which is not consistent with our product of [tex]\(-1\)[/tex].
- Option D:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_1}{x_3 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \][/tex]
- The first condition is incorrect as it doesn’t equate the slopes of opposite sides correctly.
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
1. Properties of a Rectangle:
- A rectangle is a type of parallelogram where each angle is [tex]\(90^\circ\)[/tex].
- For each internal angle to be [tex]\(90^\circ\)[/tex], any two adjacent sides must be perpendicular to each other.
2. Properties of Slopes:
- The slope of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
- Two lines are perpendicular if and only if the product of their slopes is [tex]\(-1\)[/tex].
Given these properties, let’s analyze the slopes of the given sides in the parallelogram [tex]\(ABCD\)[/tex]:
For parallelogram [tex]\(ABCD\)[/tex] to be a rectangle:
- The slopes of opposite sides must be equal.
- Adjacent sides must be perpendicular to each other.
We will denote the following slopes:
- Slope of [tex]\(AD\)[/tex]: [tex]\(\frac{y_4 - y_3}{x_4 - x_3}\)[/tex]
- Slope of [tex]\(AB\)[/tex]: [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex]
Then, for [tex]\(ABCD\)[/tex] to be a rectangle, two conditions must hold:
1. Opposite sides must have equal slopes. Hence, [tex]\( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex].
2. Adjacent sides must be perpendicular, which means [tex]\( \left( \frac{y_4 - y_3}{x_4 - x_3} \right) \times \left( \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \)[/tex].
Now, let’s look at each of the provided options:
- Option A:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_2}{x_3 - x_2} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
- This option contains an incorrect duplicate [tex]\( \frac{y_2 - y_2}{x_3 - x_2} \)[/tex], which simplifies incorrectly.
- Option B:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \][/tex]
- This option exactly matches our conditions: equal slopes of opposite sides and perpendicular adjacent sides.
- Option C:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_{1}}{x_2 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
- The second condition involves a different slope which is not consistent with our product of [tex]\(-1\)[/tex].
- Option D:
[tex]\[ \left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_1}{x_3 - x_1} \right) \text{ and } \left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1 \][/tex]
- The first condition is incorrect as it doesn’t equate the slopes of opposite sides correctly.
Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]
