PQRS is a rhombus.Its diagonals PR and QS intersect at the pint M and satisfy QS= 2 PR. If its coordinates of S andM are(1,1) and (2,-1) respectively,find the coordinates of P ?
Answers
Its diagonals PR and QS intersect at the point M and ... If the co- ordinates of S and M are (1,1) and (2,-1) ... Find the co-ordinates of P ... putting coordinates of M. 1 ...
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PQRS is a rhombus and its diagonals PR and SQ intersect at M and satisfy QS=2PR.If the coordinates of S and M are (1,1),(2,1) respectively find the coordinates of P
coordinates of S (1,1) and M (2,1)
Length of SM = √((2-1)² + (1-1)²) = √1 + 0 = 1
SM = 1
QS = 2*SM = 2 as diagonal of rhombus bisect each other perpendicularly
QS = 2PR
=> 2 = 2PR
=> PR = 1
=> PM = PR/2 = 1/2
SM =1 PM = 1/2
PS² = SM² + PM²
=> PS² = 1 + 1/4
=> PS² = 5/4
Let say co-ordinates of P (x,y) & S (1,1)
PS² = (x-1)² + (y-1)²
(x-1)² + (y-1)² = 5/4 eq1
Let say co-ordinates of P (x,y) & M (2,1)
PM² = (x-2)² + (y-1)²
(x-2)² + (y-1)² = 1/4 eq2
eq 2 - eq 1
=> (x-2)² - (x-1)² = 1/4 - 5/4
using a² - b² = (a +b)(a-b)
=> (x - 2 + x -1) (x -2 -x +1) = -1
=> (2x -3)(-1) = -1
dividing by -1 both sides
=> 2x -3 = 1
=> 2x = 4
=> x = 2
putting x =1 in (x-2)² + (y-1)² = 1/4
(2-2)² + (y-1)² = 1/4
=> (y-1)² = 1/4
square rooting both sides
=> y - 1 = ±1/2
=> y = 1/2 or 3/2
Co-ordinates of P ( 2 , 1/2) or (2 , 3/2)