Question

Explain how you would use analytic geometry to show that a quadrilateral is a parallelogram, if you are given the coordinates of the vertices.

Answer 1

To prove a quadrilateral is a parallelogram using analytic geometry, calculate and compare the slopes and lengths of opposite sides for equality, and verify if diagonals bisect by checking if midpoints coincide.

Step-by-step explanation:

To show that a quadrilateral is a parallelogram using analytic geometry when given the coordinates of its vertices, you would perform the following steps:

Calculate the slopes of opposite sides of the quadrilateral. In a parallelogram, opposite sides are parallel, which means they have the same slope.

Check the lengths of both pairs of opposite sides. If the quadrilateral is a parallelogram, the lengths of opposite sides should be equal. This can be done using the distance formula: √((x2-x1)² + (y2-y1)²).

As an additional check, calculate the midpoints of both diagonals. In a parallelogram, the diagonals bisect each other, which means the midpoints should coincide.

If both the slopes and lengths of opposite sides are equal and the midpoints of the diagonals are the same, you can conclude that the quadrilateral is a parallelogram.

Answer 2

To determine if a quadrilateral is a parallelogram using analytic geometry, you can follow these steps:

1. Given the coordinates of the vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) of the quadrilateral.

2. Calculate the slopes of the two opposite sides of the quadrilateral using the formula: slope = (y₂ - y₁) / (x₂ - x₁).

3. If the slopes of the opposite sides are equal, it indicates that the opposite sides are parallel.

4. Similarly, calculate the slopes of the other two opposite sides of the quadrilateral.

5. If the slopes of these two opposite sides are also equal, it confirms that all the sides of the quadrilateral are parallel.

6. Additionally, you can check if the opposite sides are equal in length by calculating the distance between the corresponding endpoints using the distance formula: distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²). If the distances of the opposite sides are equal, it further supports the parallelogram condition.

By performing these calculations and comparisons, you can determine if the given quadrilateral is a parallelogram using analytic geometry.