Answer :
To determine which conditions must hold true for a given parallelogram [tex]\(ABCD\)[/tex] with vertices [tex]\(A(x_1, y_1)\)[/tex], [tex]\(B(x_2, y_2)\)[/tex], [tex]\(C(x_3, y_3)\)[/tex], and [tex]\(D(x_4, y_4)\)[/tex] to be a rectangle, we need to consider the properties of a rectangle.
A rectangle is a type of parallelogram with additional properties:
1. Opposite sides are equal in length and parallel.
2. All interior angles are right angles (90 degrees).
The second property implies that the adjacent sides must be perpendicular to each other, meaning their slopes multiply to [tex]\(-1\)[/tex].
Let's calculate the slopes of the sides:
- The slope of side [tex]\(AB\)[/tex] is [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
- The slope of side [tex]\(BC\)[/tex] is [tex]\(\frac{y_3 - y_2}{x_3 - x_2}\)[/tex].
- The slope of side [tex]\(CD\)[/tex] is [tex]\(\frac{y_4 - y_3}{x_4 - x_3}\)[/tex].
- The slope of side [tex]\(DA\)[/tex] is [tex]\(\frac{y_1 - y_4}{x_1 - x_4}\)[/tex].
For [tex]\(ABCD\)[/tex] to be a rectangle:
1. The slopes of adjacent sides must be negative reciprocals. Therefore, the product of the slopes of [tex]\(AB\)[/tex] and [tex]\(BC\)[/tex] must be [tex]\(-1\)[/tex].
This can be written mathematically as:
[tex]\[ \left( \frac{y_2 - y_1}{x_2 - x_1} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
Another set of adjacent slopes is [tex]\(BC\)[/tex] and [tex]\(CD\)[/tex]:
[tex]\[ \left( \frac{y_3 - y_2}{x_3 - x_2} \times \frac{y_4 - y_3}{x_4 - x_3} \right) = -1 \][/tex]
Given multiple-choice answers, let's eliminate the incorrect ones:
(A) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_2}{x_3 - x_2} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1\)[/tex].
- This cannot be true because the slopes of opposite sides cannot be equal for a rectangle; they must be parallel.
(B) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1\)[/tex].
- This is the correct criteria as it states that the slopes of opposite sides are equal (a requirement for parallelograms) and adjacent sides are perpendicular.
(C) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1\)[/tex].
- Incorrect, similar to option B but mixes slopes improperly.
(D) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_1}{x_3 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1\)[/tex].
- Incorrect, slopes seem mismatched and do not provide parallel opposite sides.
The correct answer is:
[tex]\[ \boxed{B} \][/tex]
A rectangle is a type of parallelogram with additional properties:
1. Opposite sides are equal in length and parallel.
2. All interior angles are right angles (90 degrees).
The second property implies that the adjacent sides must be perpendicular to each other, meaning their slopes multiply to [tex]\(-1\)[/tex].
Let's calculate the slopes of the sides:
- The slope of side [tex]\(AB\)[/tex] is [tex]\(\frac{y_2 - y_1}{x_2 - x_1}\)[/tex].
- The slope of side [tex]\(BC\)[/tex] is [tex]\(\frac{y_3 - y_2}{x_3 - x_2}\)[/tex].
- The slope of side [tex]\(CD\)[/tex] is [tex]\(\frac{y_4 - y_3}{x_4 - x_3}\)[/tex].
- The slope of side [tex]\(DA\)[/tex] is [tex]\(\frac{y_1 - y_4}{x_1 - x_4}\)[/tex].
For [tex]\(ABCD\)[/tex] to be a rectangle:
1. The slopes of adjacent sides must be negative reciprocals. Therefore, the product of the slopes of [tex]\(AB\)[/tex] and [tex]\(BC\)[/tex] must be [tex]\(-1\)[/tex].
This can be written mathematically as:
[tex]\[ \left( \frac{y_2 - y_1}{x_2 - x_1} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1 \][/tex]
Another set of adjacent slopes is [tex]\(BC\)[/tex] and [tex]\(CD\)[/tex]:
[tex]\[ \left( \frac{y_3 - y_2}{x_3 - x_2} \times \frac{y_4 - y_3}{x_4 - x_3} \right) = -1 \][/tex]
Given multiple-choice answers, let's eliminate the incorrect ones:
(A) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_2}{x_3 - x_2} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1\)[/tex].
- This cannot be true because the slopes of opposite sides cannot be equal for a rectangle; they must be parallel.
(B) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1\)[/tex].
- This is the correct criteria as it states that the slopes of opposite sides are equal (a requirement for parallelograms) and adjacent sides are perpendicular.
(C) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_2 - y_1}{x_2 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_3 - y_2}{x_3 - x_2} \right) = -1\)[/tex].
- Incorrect, similar to option B but mixes slopes improperly.
(D) [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} = \frac{y_3 - y_1}{x_3 - x_1} \right)\)[/tex] and [tex]\(\left( \frac{y_4 - y_3}{x_4 - x_3} \times \frac{y_2 - y_1}{x_2 - x_1} \right) = -1\)[/tex].
- Incorrect, slopes seem mismatched and do not provide parallel opposite sides.
The correct answer is:
[tex]\[ \boxed{B} \][/tex]
