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. 

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.

Answer 2

To determine the ratio of \( a \) and \( b \) in a parallelogram with vertices \((-3,-1),(a,b),(3,3),\) and \((4,3)\), we use the property that opposite sides of a parallelogram are parallel and equal in length.

Consider the vertices in sequence as \( A(-3,-1), B(a,b), C(3,3), D(4,3) \).

For \( AB \parallel CD \):

\[\text{slope of } AB = \text{slope of } CD\]

\[\frac{b-(-1)}{a-(-3)} = \frac{3-3}{4-3}\]

\[\frac{b+1}{a+3} = 0\]

So, \( b+1 = 0 \), therefore \( b = -1 \).

For \( AD \parallel BC \):

\[\text{slope of } AD = \text{slope of } BC\]

\[\frac{3-(-1)}{4-(-3)} = \frac{3-b}{3-a}\]

\[\frac{4}{7} = \frac{3+1}{3-a}\]

\[\frac{4}{7} = \frac{4}{3-a}\]

Thus, \( 3-a = 7 \), so \( a = -4 \).

The ratio of \( a \) to \( b \) is \(-4:-1\), which simplifies to \(4:1\).

Therefore, the correct ratio of \( a \) and \( b \) is 4:1.