Question

If four vertices of a parallelogram are (-3,-1),(a,b),(3,3) and (4,3) taken in order,then the ratio of a and b is

• A. 4 : 1
• B. 1 : 2
• C. 1 : 3
• D. 3 : 1

Answer 1

The Correct Option is A

Given:

The vertices of a parallelogram are (-3, -1), (a, b), (3, 3), and (4, 3), taken in order. 

Goal: Find the ratio of a and b.

Step 1: Use the property of the midpoint of diagonals.

In a parallelogram, the diagonals bisect each other. Therefore, the midpoints of both diagonals are the same.

The midpoint of the diagonal joining (-3, -1) and (3, 3) can be calculated as:

Midpoint = \(\left(\frac{-3 + 3}{2}, \frac{-1 + 3}{2}\right) = (0, 1)\)

The midpoint of the diagonal joining (a, b) and (4, 3) is:

Midpoint = \(\left(\frac{a + 4}{2}, \frac{b + 3}{2}\right)\)

Step 2: Set the midpoints equal to each other.

Since the midpoints of the diagonals are the same, we can equate the two midpoints:

\(\left(\frac{a + 4}{2}, \frac{b + 3}{2}\right) = (0, 1)\)

Step 3: Solve for a and b.

From the x-coordinates: \(\frac{a + 4}{2} = 0\), which gives \(a + 4 = 0\), so \(a = -4\).

From the y-coordinates: \(\frac{b + 3}{2} = 1\), which gives \(b + 3 = 2\), so \(b = -1\).

Step 4: Find the ratio of a and b.

The ratio of a and b is:

\(\frac{a}{b} = \frac{-4}{-1} = 4\)

Answer: The ratio of a and b is 4 : 1.