The vertices of a parallelogram are A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4). Which of the following must be true if parallelogram ABCD is proven to be a rectangle?
To determine the properties that must be satisfied for the parallelogram ABCD to be proven as a rectangle, we need to understand the characteristics of a rectangle.
A rectangle is a special type of parallelogram that has several distinct properties:
1. Opposite sides of a rectangle are parallel and congruent.
2. All interior angles of a rectangle are right angles (90 degrees).
3. The diagonals of a rectangle are equal in length and bisect each other.
To determine which conditions must be true for the given parallelogram ABCD to be a rectangle, we can analyze the properties:
1. Opposite sides of a parallelogram ABCD are already proven to be parallel since it is mentioned as a parallelogram. This condition is already satisfied.
2. To prove that all interior angles of the parallelogram ABCD are right angles, we can examine the slopes of the lines formed by connecting the vertices.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:
slope = (y2 - y1) / (x2 - x1)
In a rectangle, the slopes of opposite sides are negative reciprocals of each other, i.e., if the slope of one side is m, the slope of the opposite side is -1/m.
Calculate the slopes of the sides AB, BC, CD, and DA using the given coordinates. Then check if the slopes of opposite sides are negative reciprocals of each other:
- Slope of AB = (y2 - y1) / (x2 - x1)
- Slope of BC = (y3 - y2) / (x3 - x2)
- Slope of CD = (y4 - y3) / (x4 - x3)
- Slope of DA = (y1 - y4) / (x1 - x4)
If the slopes of AB and CD are negative reciprocals of BC and DA, respectively, then the condition of right angles for the interior angles is satisfied.
3. To prove that the diagonals of the parallelogram ABCD are equal in length and bisect each other, we need to calculate the lengths of both diagonals using the distance formula:
The length of a diagonal between two points (x1, y1) and (x2, y2) can be calculated using the formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Calculate the lengths of both diagonals AC and BD using the given coordinates. If the lengths of AC and BD are equal and their midpoints coincide, then the condition of equal diagonals that bisect each other is satisfied.
By checking the slopes of opposite sides, the lengths of the diagonals, and the congruence of opposite sides, we can determine which conditions must be true for the parallelogram ABCD to be proven as a rectangle.
If parallelogram ABCD is proven to be a rectangle, then the following must be true:
1. Opposite sides of the parallelogram are congruent: The distance between points A and B is equal to the distance between points C and D, and the distance between points A and D is equal to the distance between points B and C.
2. Diagonals are congruent: The distance between points A and C is equal to the distance between points B and D.
3. Angles are right angles: The angle between sides AB and BC is a right angle, and the angle between sides BC and CD is also a right angle.
Therefore, if all of these conditions are met, parallelogram ABCD can be proven to be a rectangle.