# Abcd is 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. - Mathematics

ABCD is 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.

#### Solution

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

Concept: Coordinate Geometry
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#### APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 16 Coordinate Geomentry
Q 33