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.
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