ABCD is a parallelogram with vertices A (x₁ , y₁), B (x₂ , y₂) and C (x₃ , y₃). Find the coordinates of the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂ and y₃

Solution:

Given, ABCD is a parallelogram with vertices A (x₁ , y₁), B (x₂ , y₂) and C (x₃ , y₃).

We have to find the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂ and y₃.

We know that the diagonals of the parallelogram bisect each other.

So, midpoint of AC = midpoint of BD

The coordinates of the mid-point of the line segment joining the points P (x₁ , y₁) and Q (x₂ , y₂) are [(x₁ + x₂)/2, (y₁ + y₂)/2]

Midpoint of A(x₁ , y₁) and C(x₃ , y₃) = [(x₁ + x₃)/2, (y₁ + y₃)/2]

Let the coordinates of D be (x, y).

Midpoint of B(x₂ , y₂) and D(x, y) = [(x₂ + x)/2, (y₂ + y)/2]

Now, [(x₁ + x₃)/2, (y₁ + y₃)/2] = [(x₂ + x)/2, (y₂ + y)/2]

So, (x₁ + x₃)/2 = (x₂ + x)/2

Canceling out common terms,

x₁ + x₃ = x₂ + x

So, x = x₁ + x₃ - x₂

Also, (y₁ + y₃)/2 = (y₂ + y)/2

Canceling out common terms,

y₁ + y₃ = y₂ + x

So, y = y₁ + y₃ - y₂

Therefore, the coordinates of D are (x₁ + x₃ - x₂, y₁ + y₃ - y₂).

✦ Try This: The three vertices of a parallelogram taken in order are (-1, 0), (3, 1) and (2, 2) respectively.Find the coordinates of the fourth vertex.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Sample Problem 4

ABCD is a parallelogram with vertices A (x₁ , y₁), B (x₂ , y₂) and C (x₃ , y₃). Find the coordinates of the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂ and y₃

Summary:

ABCD is a parallelogram with vertices A (x₁ , y₁), B (x₂ , y₂) and C (x₃ , y₃). The coordinates of the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂ and y₃ are (x₁ + x₃ - x₂, y₁ + y₃ - y₂)

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