If the points A (1, –2), B (2, 3) C (a, 2) and D (– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as base
Solution:
Given, the points A(1, -2) B(2, 3) C(a, 2) and D(-4, -3) form a parallelogram.
We have to find the value of a and the height of the parallelogram taking AB as base.
We know that the diagonals of a parallelogram bisect each other.
So, the diagonal AC and BD of the parallelogram ABCD bisect each other.
i.e., 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(1, -2) and C(a, 2) = [(1 + a)/2, (-2 + 2)/2]
= [(1 + a)/2, 0]
Midpoint of B(2, 3) and D(-4, -3) = [(2 - 4)/2, (3 - 3)/2]
= [(-2/2), 0]
= [-1, 0]
Now, [(1 + a)/2, 0] = (-1, 0)
So, 1 + a/2 = -1
1 + a = -2
a = -2 - 1
a = -3
Now, the area of parallelogram ABCD = 2(area of triangle ABC)
The area of a triangle with vertices A (x₁ , y₁) , B (x₂ , y₂) and C (x₃ , y₃) is
1/2[x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)
Here, (x₁, y₁) = (1, -2) (x₂, y₂) = (2, 3) and (x₃, y₃) = (-3, 2)
Now, the area of triangle ABC = 1/2[1(3 - 2) + 2(2 - (-2) + -3(-2 - 3)]
= 1/2[(1) + 2(4) - 3(-5)]
= 1/2[1 + 8 + 15]
= 1/2[24]
= 12 square units
Area of parallelogram = 2(12) = 24 square units
Base = AB
The distance between two points P (x₁ , y₁) and Q (x₂ , y₂) is
√[(x₂ - x₁)² + (y₂ - y₁)²]
Distance between A(1, -2) and B(2, 3) = √[(2 - 1)² + (3 - (-2))²]
= √[(1)² + (5)²]
= √(1 + 25)
= √26
We know that area of parallelogram = base × height
24 = √26 × height
Height = 24/√26
Therefore, the height of the parallelogram is 24/√26 units.
✦ Try This: If the points A (1, -2), B (2, 3) C (k, 2) and D (-1, -3) form a parallelogram, find the value of k and height of the parallelogram taking AB as base.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7
NCERT Exemplar Class 10 Maths Exercise 7.4 Problem 4
If the points A (1, –2), B (2, 3) C (a, 2) and D (– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as base
Summary:
If the points A (1, –2), B (2, 3) C (a, 2) and D (– 4, –3) form a parallelogram, the value of a is -3 and height of the parallelogram taking AB as base is 24/√26units
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