precise name of quadrilateral with vertices A(-2,4) B(2,6) C(6,4) D(2,-3)

To find the precise name of the quadrilateral with the given vertices A(-2,4), B(2,6), C(6,4), D(2,-3), we need to analyze its properties.

1. Check for Parallel Sides:
We can start by checking if any pair of opposite sides are parallel. To do this, we need to calculate the slopes of the opposite sides.

Slope of AB = (6 - 4) / (2 - (-2)) = 2 / 4 = 1/2
Slope of BC = (4 - 6) / (6 - 2) = -2 / 4 = -1/2
Slope of CD = (-3 - 6) / (2 - 2) = -9 / 0 (undefined)

From the slopes, we see that AB and BC have slopes of opposite signs and reciprocals of each other, which means they are parallel. However, CD has an undefined slope, indicating it is a vertical line. Hence, CD is not parallel to AB and BC.

2. Check for Equal Side Lengths:
Next, we check if the lengths of the opposite sides are equal.

Distance between A and B = √[(2 - (-2))^2 + (6 - 4)^2] = √[4 + 4] = √8 = 2√2
Distance between B and C = √[(6 - 2)^2 + (4 - 6)^2] = √[16 + 4] = √20 = 2√5
Distance between C and D = √[(2 - 6)^2 + (-3 - 4)^2] = √[16 + 49] = √65

The opposite sides AB and CD have different lengths, so the quadrilateral is not a parallelogram.

3. Check for Right Angles:
We can also check if any angle in the quadrilateral is a right angle. To do this, we calculate the slopes of the adjacent sides and check if they are negative reciprocals.

Slope of AB = 1/2
Slope of BC = -1/2
Slope of CD = undefined

From the slopes, we see that AB and BC have negative reciprocal slopes, indicating that angle ABC is a right angle.

4. Check for Additional Properties:
Considering the given properties above, the quadrilateral can be identified as:

- ABCD is NOT a parallelogram because opposite sides are not parallel.
- ABCD is NOT a rectangle because opposite sides are not parallel and adjacent sides do not have equal lengths.
- ABCD is NOT a rhombus because opposite sides do not have equal lengths.
- ABCD is NOT a square because opposite sides are not parallel and adjacent sides do not have equal lengths.

Hence, based on the given information, the quadrilateral can be simply referred to as a "quadrilateral ABCD" since it does not meet the criteria to be named as a specific type of quadrilateral.