ABCD is a rectangle formed by the points A(-1, -1), B(-1, 4), C(5, 4) and D(5, -1). P, Q, R, S are the mid points of AB, BC, CD, DA. Is the Quadrilateral a square? A rectangle? Or a rhombus? Justify your answer.

To determine the nature of the quadrilateral formed by the midpoints P, Q, R, and S of the rectangle ABCD, we first need to find the coordinates of these midpoints.

Finding the Midpoints

• P: Midpoint of AB = ((-1 + -1)/2, (-1 + 4)/2) = (-1, 1.5)
• Q: Midpoint of BC = ((-1 + 5)/2, (4 + 4)/2) = (2, 4)
• R: Midpoint of CD = ((5 + 5)/2, (4 + -1)/2) = (5, 1.5)
• S: Midpoint of DA = ((5 + -1)/2, (-1 + -1)/2) = (2, -1)

Coordinates of Midpoints

The coordinates of the midpoints are:

• P(-1, 1.5)
• Q(2, 4)
• R(5, 1.5)
• S(2, -1)

Analyzing the Quadrilateral PQRS

Next, we need to check the lengths of the sides and the diagonals of quadrilateral PQRS to classify it.

Calculating Side Lengths

• Length PQ = √[(2 - (-1))² + (4 - 1.5)²] = √[3² + 2.5²] = √[9 + 6.25] = √15.25
• Length QR = √[(5 - 2)² + (1.5 - 4)²] = √[3² + (-2.5)²] = √[9 + 6.25] = √15.25
• Length RS = √[(2 - 5)² + (-1 - 1.5)²] = √[(-3)² + (-2.5)²] = √[9 + 6.25] = √15.25
• Length SP = √[(-1 - 2)² + (1.5 - (-1))²] = √[(-3)² + (2.5)²] = √[9 + 6.25] = √15.25

Checking the Diagonals

• Diagonal PR = √[(5 - (-1))² + (1.5 - 1.5)²] = √[6² + 0²] = 6
• Diagonal QS = √[(2 - 2)² + (-1 - 4)²] = √[0² + (-5)²] = 5

Conclusion on the Shape

Since all sides of quadrilateral PQRS are equal (√15.25) and the diagonals are not equal (6 and 5), this shape is classified as a rhombus. It is not a square because the angles are not right angles, and it is not a rectangle because the sides are not equal.