Question

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

Answer 1

Step-by-step explanation:

can find co-ordinate pqrs by midpoint formula

Answer 2

PQRS is a quadrilateral, not a square, rectangle or rhombus.

Step-by-step explanation:

Given: A (-1, -1), B (-1, 4), C (5, 4) and D (5,-1). P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively.  

Solution:

Coordinates of P = (-1-1/2 , -1+4/2) = (-1, 3/2)

Q = (2, 0)

R = (0, 3/2)

S = (2, -1)

d(P, Q) = √ (x2 − x1)² + (y2 − y1)² = √(2+1)² + (-3/2)² = √ 9 +9/4 = √45/4 = 3√5/2

d(Q, R) = √ (-2)² + (3/2)² = √ 4 + 9/4 = √25/4 = 5/2

d(R, S) = √ (2)² + (-1-3/2)² = √4 + (-5/2)² = √4 + 25/4 = (√41)/2

d(S, P) = √ (3)² + (-1-3/2)² = √9 + (-5/2)² = √61/4

We find that the sides are of different lengths. So PQRS is a quadrilateral, not a square, rectangle or rhombus.