To find the value of y such that the points P(2, 4), Q(0, 3), R(3, 6), and S(5, y) form a parallelogram PQRS, we use the property that the diagonals of a parallelogram bisect each other. This means the midpoints of the diagonals PR and QS must coincide.

 Step 1: Find the midpoint of diagonal PR
The formula for the midpoint of a line segment joining two points (x₁, y₁) and (x₂, y₂) is:

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

For diagonal PR, the endpoints are P(2, 4) and R(3, 6). The midpoint of PR is:

Midpoint of PR = ((2 + 3)/2, (4 + 6)/2)
= (5/2, 10/2)
= (5/2, 5)

Step 2: Find the midpoint of diagonal QS
For diagonal QS, the endpoints are Q(0, 3) and S(5, y). The midpoint of QS is:

Midpoint of QS = ((0 + 5)/2, (3 + y)/2)
= (5/2, (3 + y)/2)

Step 3: Equate the midpoints
Since the diagonals of a parallelogram bisect each other, the midpoints of PR and QS must be equal. Therefore:

(5/2, 5) = (5/2, (3 + y)/2)

Equating the y-coordinates:

5 = (3 + y)/2

Multiply through by 2 to eliminate the denominator:

10 = 3 + y

Solve for y:

y = 10 – 3
y = 7

 Step 4: Verify the solution
Substitute y = 7 into the coordinates of S(5, y), making S(5, 7). Recalculate the midpoint of QS:

Midpoint of QS = ((0 + 5)/2, (3 + 7)/2)
= (5/2, 10/2)
= (5/2, 5)

This matches the midpoint of PR, confirming that the diagonals bisect each other.

The correct answer is:
a) 7
This question related to Chapter 7 Mathematics Class 10th NCERT. From the Chapter 7 Coordinate Geometry. Give answer according to your understanding.

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