If the three vertices of the parallelogram ABCD are.A(1, a), B(3, a), C(2, b), then D is equal toa)(3,b)b)(6,b)c)(4,b)d)(0,b)Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Given:
The coordinates of three vertices of the parallelogram ABCD are A(1, a), B(3, a), C(2, b).

To find:
The coordinates of vertex D.

Solution:
Step 1:
Since ABCD is a parallelogram, opposite sides are parallel. Therefore, the slope of AB is equal to the slope of CD, and the slope of BC is equal to the slope of AD.

Step 2:
Find the slope of AB:
m_AB = (a - a) / (3 - 1) = 0 / 2 = 0

Step 3:
Find the slope of BC:
m_BC = (b - a) / (2 - 3) = (b - a) / (-1)

Step 4:
Since AB and CD are parallel, the slope of CD is also 0.

Step 5:
Using the slope-intercept form of a line, we can find the equation of CD:
y - b = 0(x - x₁), where x₁ = 2 and y₁ = b
y - b = 0(x - 2)
y - b = 0
y = b

Step 6:
Since BC and AD are parallel, their slopes are equal:
(b - a) / (-1) = (y - a) / (x - 1)

Step 7:
Substitute the equation of CD into the above equation:
(b - a) / (-1) = (b - a) / (x - 1)

Step 8:
Cross-multiply:
-1(b - a) = (b - a)(x - 1)

Step 9:
Cancel out (b - a) on both sides:
-1 = x - 1

Step 10:
Simplify:
x = 0

Step 11:
Therefore, the x-coordinate of vertex D is 0.

Step 12:
Now, substitute x = 0 into the equation of CD:
y = b

Step 13:
Therefore, the y-coordinate of vertex D is b.

Conclusion:
The coordinates of vertex D are (0, b), which corresponds to option (d) in the given options.

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