the equation of two adjacent sides of rhombus are given by `y=x` and `y=7x`. the diagonals of the rhombus intersect each other at point of `(1,2)`.then the area of the rhombus is:

Options

A. `(10)/(3)`
B. `(20)/(3)`
C. `(40)/(3)`
D. `(50)/(3)`

Answer

To find the area of the rhombus given the equations of two adjacent sides and the intersection point of the diagonals, we can follow these steps:

### Step 1: Identify the equations of the sides
The equations of the two adjacent sides of the rhombus are:
1. \( y = x \)
2. \( y = 7x \)

### Step 2: Find the intersection points of the sides with the diagonals
The diagonals intersect at the point \( (1, 2) \). We will denote the vertices of the rhombus as \( A, B, C, D \).

### Step 3: Find the coordinates of the vertices
Since the diagonals bisect each other at the point \( (1, 2) \), we can find the coordinates of the vertices \( B \) and \( D \) using the midpoint theorem.

Let \( C \) be the vertex on the line \( y = x \) and \( D \) be the vertex on the line \( y = 7x \).

Using the midpoint theorem:
- For point \( C \) on line \( y = x \):
\[ \text{Let } C = (x_1, x_1) \text{ (since } y = x\text{)} \]
The midpoint condition gives us:
\[ \frac{x_1 + 0}{2} = 1 \quad \Rightarrow \quad x_1 = 2 \]
\[ \frac{x_1 + 0}{2} = 2 \quad \Rightarrow \quad x_1 = 4 \]
Thus, \( C = (2, 2) \).

- For point \( D \) on line \( y = 7x \):
\[ \text{Let } D = (x_2, 7x_2) \]
The midpoint condition gives us:
\[ \frac{x_2 + 0}{2} = 1 \quad \Rightarrow \quad x_2 = 2 \]
\[ \frac{7x_2 + 0}{2} = 2 \quad \Rightarrow \quad 7x_2 = 4 \quad \Rightarrow \quad x_2 = \frac{4}{7} \]
Thus, \( D = \left(\frac{4}{7}, 4\right) \).

### Step 4: Find the lengths of the diagonals
- Diagonal \( AC \):
\[ A = (0, 0), C = (2, 2) \]
The length of diagonal \( AC \) is:
\[ AC = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

- Diagonal \( BD \):
\[ B = \left(\frac{1}{3}, \frac{7}{3}\right), D = \left(\frac{4}{7}, 4\right) \]
The length of diagonal \( BD \) is:
\[ BD = \sqrt{\left(\frac{1}{3} - \frac{4}{7}\right)^2 + \left(\frac{7}{3} - 4\right)^2} \]
Calculate the x-coordinates:
\[ \frac{1}{3} - \frac{4}{7} = \frac{7 - 12}{21} = -\frac{5}{21} \]
Calculate the y-coordinates:
\[ \frac{7}{3} - 4 = \frac{7 - 12}{3} = -\frac{5}{3} \]
Thus,
\[ BD = \sqrt{\left(-\frac{5}{21}\right)^2 + \left(-\frac{5}{3}\right)^2} = \sqrt{\frac{25}{441} + \frac{25}{9}} = \sqrt{\frac{25}{441} + \frac{1225}{441}} = \sqrt{\frac{1250}{441}} = \frac{5\sqrt{50}}{21} = \frac{25\sqrt{2}}{21} \]

### Step 5: Calculate the area of the rhombus
The area \( A \) of the rhombus is given by:
\[ A = \frac{1}{2} \times AC \times BD \]
Substituting the lengths of the diagonals:
\[ A = \frac{1}{2} \times (2\sqrt{2}) \times \left(\frac{25\sqrt{2}}{21}\right) \]
\[ A = \frac{25 \times 2}{42} = \frac{50}{42} = \frac{25}{21} \]

Thus, the area of the rhombus is \( \frac{25}{21} \) square units.