Answer :
To determine if a parallelogram on a coordinate grid is a rectangle, we need to confirm two key characteristics of rectangles:
1. Opposite sides are equal in length.
2. Adjacent sides are perpendicular to each other.
Now, let's look at the formulas listed and determine which are applicable to these characteristics:
1. Distance Formula: This allows you to calculate the distance between two points on a coordinate grid. Since a rectangle has its opposite sides equal in length, we can use the Distance Formula to verify this. An example of the Distance Formula is given by,
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of the points.
2. Midpoint Formula: This formula finds the midpoint of the line segment connecting two points on a coordinate plane. This isn't directly relevant to determining whether we have a rectangle or not since knowing the midpoint doesn't tell us about side lengths or angles.
3. Slope Formula: This is used to find the slope of a line on a coordinate grid. For a parallelogram to be a rectangle, each pair of adjacent sides must be perpendicular. Two lines are perpendicular if the product of their slopes is -1. Thus, we can use the Slope Formula to verify this second characteristic of a rectangle. The Slope Formula is,
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of two points on the line.
4. Quadratic Formula: This is generally used to find the solutions to a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. This formula does not help in determining the properties of a parallelogram or proving that it is a rectangle.
To summarize, we use:
- The Distance Formula to confirm that opposite sides are equal in length.
- The Slope Formula to confirm that adjacent sides meet at right angles (or, more technically, are perpendicular to one another).
Therefore, to determine if a parallelogram is a rectangle, you would use the Distance Formula and the Slope Formula.
1. Opposite sides are equal in length.
2. Adjacent sides are perpendicular to each other.
Now, let's look at the formulas listed and determine which are applicable to these characteristics:
1. Distance Formula: This allows you to calculate the distance between two points on a coordinate grid. Since a rectangle has its opposite sides equal in length, we can use the Distance Formula to verify this. An example of the Distance Formula is given by,
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of the points.
2. Midpoint Formula: This formula finds the midpoint of the line segment connecting two points on a coordinate plane. This isn't directly relevant to determining whether we have a rectangle or not since knowing the midpoint doesn't tell us about side lengths or angles.
3. Slope Formula: This is used to find the slope of a line on a coordinate grid. For a parallelogram to be a rectangle, each pair of adjacent sides must be perpendicular. Two lines are perpendicular if the product of their slopes is -1. Thus, we can use the Slope Formula to verify this second characteristic of a rectangle. The Slope Formula is,
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of two points on the line.
4. Quadratic Formula: This is generally used to find the solutions to a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. This formula does not help in determining the properties of a parallelogram or proving that it is a rectangle.
To summarize, we use:
- The Distance Formula to confirm that opposite sides are equal in length.
- The Slope Formula to confirm that adjacent sides meet at right angles (or, more technically, are perpendicular to one another).
Therefore, to determine if a parallelogram is a rectangle, you would use the Distance Formula and the Slope Formula.
