Question

Determine whether the parallelogram is a rhombus, rectangle, square, or none. Explain.

Q(1, 3), R(3, 4), S(5, 3), T(3, 2)


Select one:

A. QRST is a rhombus that is not a square because its diagonals are perpendicular but not congruent.

B. QRST is a rectangle that is not a square because its diagonals are congruent but not perpendicular.

C. QRST is a square because its diagonals are both perpendicular and congruent.

D. QRST is none of these because its diagonals are neither congruent nor perpendicular.

Answer 1

Given the coordinates Q(1, 3), R(3, 4), S(5, 3), T(3, 2), the shape formed is a rhombus because it has all sides equal and the diagonals are perpendicular but not congruent.

Step-by-step explanation:

To determine whether the given points Q(1, 3), R(3, 4), S(5, 3), T(3, 2) form a rhombus, rectangle, square, or if they do not fit any of these categories, we first need to calculate the lengths of the sides and diagonals of the parallelogram they might form.

Side QR: √((3-1)² + (4-3)²) = √(4 + 1) = √5
Side RS: √((5-3)² + (3-4)²) = √(4 + 1) = √5
Side ST: √((5-3)² + (3-2)²) = √(4 + 1) = √5
Side TQ: √((1-3)² + (2-3)²) = √(4 + 1) = √5

Since all sides are equal, then QRST could potentially be a rhombus or a square. Next, let's check the diagonals.

Diagonal QS: √((5-1)² + (3-3)²) = √(16 + 0) = 4
Diagonal RT: √((3-3)² + (4-2)²) = √(0 + 4) = 2

The diagonals are not equal, so QRST cannot be a square or a rectangle. However, since the diagonals are perpendicular (as the product of their slopes is -1), QRST is a rhombus and not a square.

The correct answer is:

• A. QRST is a rhombus that is not a square because its diagonals are perpendicular but not congruent.