Question

If P( 8,5) Q(-7,-5) and R( -5,5)are three vertices of a parallelogram PQRS in order , What is the fourth vertex?

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\bigstar{\bold{Fourth\:vertex=(10,15)}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• Point P (8, 5)
• Point Q (-7, -5)
• Point R (-5 , 5)

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Point S

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Let Point S be ( x, y )

→ Since this is a parallellogram,

  we know that diagonals bisect each other, ie

→ Here PR and QS are the diagonals of the parallelogram

→ Hence,

  Midpoint of PR = Midpoint of QS

→ First let us find the midpoint of PR

→ We know midpoint of two points (x₁ , y₁ ) and (x₂ , y₂) is given by

  Midpoint =( (x₁ + x₁)/2 , (y₁ + y₂)/2 )

→ Here x₁ = 8, x₂ = -5, y₁ = 5, y₂ = 5

→ Substituting the datas we get,

  Midpoint of PR =( ( 8 - 5)/2 , ( 5 + 5) /2 )

  Midpoint of PR = ( 3/2 , 5 )------(1)

→ Now we have to find the midpoint of QS

→ Here x₁ = -7 , x₂ = x, y₁ = -5, y₂ = y

→ Substituting the datas,

  Midpoint of QS = ( ( -7 + x) /2 , ( -5  + y) /2 ) ------(2)

→ We know that the LHS of equation 1 and 2 are equal, hence RHS must also be equal.

→ ( -7 + x ) /2 , (-5 + y)/2 = (3/2, 5)

→ Taking x and y coordinates separately,

→  (-7 + x )/2 = 3/2

→ Cancelling 2 on both sides,

  -7 + x = 3

         x = 3 + 7

         x = 10

→ Hence the x coordinate is 10.

→ ( -5 + y ) /2 = 5

   -5 + y = 10

          y = 10 + 5

          y = 15

→ Hence the y coordinate is 15

→ ∴ The point is ( x, y) = ( 10, 15)

$\boxed{\bold{Fourth\:vertex=(10,15)}}$

$\Large{\underline{\underline{\bf{Notes:}}}}$

→ The midpoint of two points is given by,

   Midpoint =( (x₁ + x₁)/2 , (y₁ + y₂)/2 )