# If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y

**Solution:**

The coordinates of the point P(x, y) which divides the line segment joining the points A (x₁, y₁) and B(x₂, y₂), internally, in the ratio m₁: m₂ is given by the section formula: P(x, y) = [(mx₂ + nx₁)_{ }/ m + n, (my₂ + ny₁)_{ }/ m + n]

Let A (1, 2), B (4, y), C(x, 6), and D (3, 5) be the vertices of a parallelogram ABCD.

Since the diagonals of a parallelogram bisect each other. The intersection point O of diagonal AC and BD also divides these diagonals in the ratio 1:1.

Therefore, O is the mid-point of AC and BD.

According to the mid point formula,

O(x, y) = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

If O is the mid-point of AC, then the coordinates of O are

[(1 + x) / 2, (2 + 6) / 2]

⇒ [(x + 1) / 2, 4] ----- (1)

If O is the mid-point of BD, then the coordinates of O are

[(4 + 3) / 2, (5 + y) / 2]

⇒ [7/2, (5 + y) / 2] ------ (2)

Since both the coordinates are of the same point O, so, (x + 1) / 2 = 7 / 2 and 4 = (5 + y) / 2 [From equation(1) and (2)]

⇒ x + 1 = 7 and 5 + y = 8 (By cross multiplying & transposing)

⇒ x = 6 and y = 3

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

**Video Solution:**

## If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

NCERT Class 10 Maths Solutions Chapter 7 Exercise 7.2 Question 6

**Summary:**

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, then x = 6, and y = 3.

**☛ Related Questions:**

- Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, - 3) and B is (1, 4).
- If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.
- Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.
- Find the area of a rhombus if its vertices are (3, 0), (4, 5), (- 1, 4) and (- 2, - 1) taken in order. [Hint: Area of a rhombus =1/2 x (product of its diagonals)]

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