Coordinate Proofs: Parallelograms and Special Quadrilaterals

Find the midpoint of segment AC where A = (-2, 1) and C = (4, 7).

Midpoint = ((-2 + 4)/2, (1 + 7)/2) = (1, 4).

Quadrilateral PQRS has vertices P(0, 0), Q(4, 2), R(5, 6), and S(1, 4). Calculate the midpoint of PR and the midpoint of QS to determine if PQRS is a parallelogram.

Midpoint of PR = (2.5, 3). Midpoint of QS = (2.5, 3). Since the midpoints are identical, PQRS is a parallelogram.

Three vertices of parallelogram ABCD are A(1, 1), B(4, 2), and C(3, 5). Find the coordinates of vertex D(x, y).

The midpoint of diagonal AC is (2, 3). For diagonal BD to have the same midpoint, (4 + x)/2 = 2 and (2 + y)/2 = 3. Solving gives x = 0 and y = 4, so D = (0, 4).

Parallelogram ABCD has vertices A(0, 0), B(3, 4), C(8, 4), and D(5, 0). Calculate the lengths of sides AB and BC to determine if ABCD is a rhombus.

Length of AB = sqrt((3 - 0)^2 + (4 - 0)^2) = 5. Length of BC = sqrt((8 - 3)^2 + (4 - 4)^2) = 5. Since adjacent sides are equal in length, ABCD is a rhombus.

Quadrilateral JKLM has vertices J(-1, 2), K(3, 5), L(0, 9), and M(-4, 6). Verify whether JKLM is a rectangle using the slopes of adjacent sides JK and KL.

Slope of JK = (5 - 2)/(3 - (-1)) = 3/4. Slope of KL = (9 - 5)/(0 - 3) = -4/3. Since (3/4) * (-4/3) = -1, adjacent sides are perpendicular, confirming JKLM has right angles and is a rectangle.

A student tests quadrilateral ABCD by showing that side AB is parallel to side CD and that side BC is congruent to side AD. Explain why this does not guarantee ABCD is a parallelogram.

Having one pair of parallel sides and a different pair of congruent sides can define an isosceles trapezoid, not necessarily a parallelogram. To prove a parallelogram, one pair must be both parallel and congruent, or both opposite pairs must be parallel.