If one side of a rhombus has endpoints (4, 5) and (1, 1), then the maximum area of the rhombus is 50 sq. units (b) 25 sq. units 30 sq. units (d) 20 sq. units

Options

A. 50 sq.units
B. 25 sq.units
C. 30 sq.units
D. 20 sq.units

Explanation

To find the maximum area of the rhombus with one side having endpoints (4, 5) and (1, 1), we can follow these steps:

### Step 1: Identify the endpoints of the side of the rhombus
Let the endpoints of one side of the rhombus be:
- Point A (4, 5)
- Point B (1, 1)

### Step 2: Calculate the length of side AB
We can use the distance formula to find the length of side AB:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ AB = \sqrt{(1 - 4)^2 + (1 - 5)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
So, the length of side AB is 5 units.

### Step 3: Set up the area formula for the rhombus
The area \( A \) of a rhombus can be expressed in terms of the base and height. If we take AB as the base, we need to find the height (h) from point D (the opposite vertex) to line AB.

### Step 4: Express the area of the rhombus
The area of the rhombus can be calculated as:
\[ \text{Area of rhombus} = 2 \times \text{Area of triangle ABD} \]
The area of triangle ABD can be calculated as:
\[ \text{Area of triangle ABD} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times h \]
Substituting the length of AB:
\[ \text{Area of triangle ABD} = \frac{1}{2} \times 5 \times h = \frac{5h}{2} \]
Thus, the area of the rhombus becomes:
\[ \text{Area of rhombus} = 2 \times \frac{5h}{2} = 5h \]

### Step 5: Maximize the area
To maximize the area of the rhombus, we need to maximize the height \( h \). The maximum height occurs when the angle between the sides of the rhombus is 90 degrees, making it a square. In this case, the maximum height is equal to the length of the side \( AB \), which is 5.

Thus, the maximum area of the rhombus is:
\[ \text{Maximum Area} = 5 \times 5 = 25 \text{ square units} \]

### Conclusion
The maximum area of the rhombus is **25 square units**.

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