Question

ABCD is a rectangle formed by the points A(–1, –1), B(–1, 4), C(5, 4) and D(5, –1). If P, Q, R and S be the midpoints of AB, BC, CD and DA respectively, show that PQRS is a rhombus.

Answer 1

Here, the points P,Q,  Rand S are the midpoint of AB,BC, CD and DA respectively. Then

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

Coordinates of `Q = ((-1+5)/2, (4+4)/2) = (2,.4)`

Coordinates of `R = ((5+5)/2, (4-1)/2)= (5,3/2)`

Coordinates of `S = ((-1+5)/2,(-1-1)/2) = (2,-1)`

Now,

`PQ = sqrt((2+1)^2 +(4-3/2)^2) = sqrt(9+25/4) = sqrt(61/2)`

`QR = sqrt((5-2)^2 +(3/2-4)^2 )= sqrt(9+25/4) = sqrt(61/2)` 

`RS = sqrt((5-2)^2 +(3/2+1)^2 )= sqrt(9+25/4) = sqrt(61/2)`

`SP = sqrt((2+1)^2 +(-1-3/2)^2 )= sqrt(9+25/4) = sqrt(61/2)`

` PR = sqrt((5-1)^2 +(3/2-3/2)^2) = sqrt(36) = 6`

`QS = sqrt((2-2)^2 +(-1-4)^2) = sqrt(25) =5`

Thus, PQ = QR = RS  = SP and PR ≠ QS  therefore PQRS is a rhombus