Classifying a Coordinate Quadrilateral
A surveyor places stakes at A(1, 2), B(5, 4), C(6, 8), and D(2, 6) to mark a proposed rectangular plaza. A draftsperson claims the plaza is a rectangle because AB and CD appear equal and parallel, and BC and DA also appear equal and parallel. The project engineer wants a precise classification before ordering edging, because the edging price depends on perimeter and the paving pattern depends on whether the corners are right angles. The available data are only the coordinates; no angles were measured in the field. The team must use slopes and distances to test the rectangle claim, compute the exact perimeter and area, and decide whether the shape is a parallelogram, a rhombus, a rectangle, or none of these. A second concern is whether the diagonals are equal, since equal diagonals are a rectangle test for parallelograms.
1. Test whether ABCD is a parallelogram. Show slope calculations and side-length comparisons.
Read the worked answer
Slope AB = (4 − 2)/(5 − 1) = 2/4 = 1/2. Slope BC = (8 − 4)/(6 − 5) = 4/1 = 4. Slope CD = (6 − 8)/(2 − 6) = (−2)/(−4) = 1/2. Slope DA = (2 − 6)/(1 − 2) = (−4)/(−1) = 4. Since AB and CD have the same slope, they are parallel; since BC and DA have the same slope, they are parallel. Both pairs of opposite sides are parallel, so ABCD is a parallelogram. Length AB = √((5−1)² + (4−2)²) = √(16 + 4) = √20 = 2√5. Length CD is the same by symmetry. Length BC = √((6−5)² + (8−4)²) = √(1 + 16) = √17, and DA = √17. Opposite sides are equal, confirming the parallelogram classification.
2. Is ABCD a rectangle? Use slopes or a dot product, and compare the diagonals.
Read the worked answer
Adjacent sides AB and BC have slopes 1/2 and 4. Their product is (1/2)(4) = 2, not −1, so the sides are not perpendicular. Equivalently, vector AB = (4, 2) and vector BC = (1, 4); their dot product is 4(1) + 2(4) = 12, not 0. Therefore the corners are not right angles, so ABCD is not a rectangle. Check the diagonals as a secondary test: AC = √((6−1)² + (8−2)²) = √(25 + 36) = √61, while BD = √((2−5)² + (6−4)²) = √(9 + 4) = √13. The diagonals are not equal, which is consistent with the shape not being a rectangle. It is also not a rhombus, because adjacent sides √20 and √17 are not equal. The precise classification is parallelogram only.
3. Compute the exact perimeter and area of ABCD. Explain which measurement the edging order and the paving order require.
Read the worked answer
Perimeter = AB + BC + CD + DA = √20 + √17 + √20 + √17 = 2√20 + 2√17 = 4√5 + 2√17, approximately 8.944 + 8.246 = 17.19 units. For area, use the cross product of adjacent side vectors AB = (4, 2) and BC = (1, 4). The area is |4(4) − 2(1)| = |16 − 2| = 14 square units. Equivalently, the shoelace formula on the four vertices also gives 14. The edging order is based on the perimeter because edging runs along the boundary; the paving order is based on the area because paving covers the interior. Neither order should use the draftsperson’s rectangle assumption, since the corners are not right angles.