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.
