The opposite angular points of a square are (3,4) and (1,-1) then the other two vertices are ……and ……….

Answer

To find the other two vertices of the square given the opposite angular points (3, 4) and (1, -1), we can follow these steps:

### Step 1: Identify the given points
Let A = (3, 4) and C = (1, -1) be the opposite vertices of the square.

### Step 2: Find the midpoint of the diagonal
The midpoint M of the diagonal AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of A and C:
\[ M = \left( \frac{3 + 1}{2}, \frac{4 + (-1)}{2} \right) = \left( \frac{4}{2}, \frac{3}{2} \right) = (2, 1.5) \]

### Step 3: Calculate the distance between A and C
Using the distance formula, the distance \(d\) between points A and C is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and C:
\[ d = \sqrt{(1 - 3)^2 + (-1 - 4)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \]

### Step 4: Find the length of the sides of the square
Since A and C are opposite vertices, the length of one side of the square \(s\) can be found using the relationship:
\[ s = \frac{d}{\sqrt{2}} = \frac{\sqrt{29}}{\sqrt{2}} = \frac{\sqrt{58}}{2} \]

### Step 5: Determine the slope of line AC
The slope \(m\) of line AC is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 4}{1 - 3} = \frac{-5}{-2} = \frac{5}{2} \]

### Step 6: Find the slopes of the other two sides
The slopes of the sides of the square will be perpendicular to the slope of AC. Therefore, the slopes of the other two sides will be:
\[ m_1 = -\frac{1}{m} = -\frac{2}{5} \]
and
\[ m_2 = -m_1 = \frac{2}{5} \]

### Step 7: Write the equations of the lines through point M
Using point-slope form \(y - y_1 = m(x - x_1)\), we can write the equations of the lines through M (2, 1.5) with slopes \(m_1\) and \(m_2\).

For slope \(m_1 = -\frac{2}{5}\):
\[ y - 1.5 = -\frac{2}{5}(x - 2) \]
Simplifying gives:
\[ y = -\frac{2}{5}x + \frac{4}{5} + 1.5 = -\frac{2}{5}x + \frac{4}{5} + \frac{7.5}{5} = -\frac{2}{5}x + \frac{11.5}{5} \]

For slope \(m_2 = \frac{2}{5}\):
\[ y - 1.5 = \frac{2}{5}(x - 2) \]
Simplifying gives:
\[ y = \frac{2}{5}x - \frac{4}{5} + 1.5 = \frac{2}{5}x - \frac{4}{5} + \frac{7.5}{5} = \frac{2}{5}x + \frac{3.5}{5} \]

### Step 8: Find the intersection points with the lines
To find the intersection points, we can substitute values into the equations derived above.

1. For \(y = -\frac{2}{5}x + \frac{11.5}{5}\):
- Substitute \(y = 1\) to find the x-coordinate.
- Solve for \(x\) to find the first vertex.

2. For \(y = \frac{2}{5}x + \frac{3.5}{5}\):
- Substitute \(y = 1\) to find the x-coordinate.
- Solve for \(x\) to find the second vertex.

### Step 9: Calculate the coordinates of the other two vertices
After solving the equations, we find the coordinates of the other two vertices of the square.

### Final Answer
The other two vertices of the square are \(\left(\frac{9}{2}, \frac{1}{2}\right)\) and \(\left(-\frac{1}{2}, \frac{5}{2}\right)\).

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