The co-ordinates of the vertex A of a square ABCD are (1, 2) and the equation of the diagonal BD is `x+2y= 10`. Find the equation of the other diagonal and the co-ordinates of the centre of the square.

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript.

### Step 1: Identify the given information
We are given:
- The coordinates of vertex A of square ABCD: \( A(1, 2) \)
- The equation of diagonal BD: \( x + 2y = 10 \)

### Step 2: Determine the slope of diagonal BD
The equation of line BD can be rewritten in slope-intercept form (y = mx + b):
\[ 2y = -x + 10 \]
\[ y = -\frac{1}{2}x + 5 \]
From this, we can see that the slope (m) of line BD is \( -\frac{1}{2} \).

### Step 3: Find the slope of diagonal AC
Since the diagonals of a square are perpendicular to each other, the slope of diagonal AC will be the negative reciprocal of the slope of diagonal BD.
- Slope of BD: \( -\frac{1}{2} \)
- Slope of AC: \( m_{AC} = 2 \) (since \( m_{AC} \cdot m_{BD} = -1 \))

### Step 4: Write the equation of diagonal AC
We use the point-slope form of the line equation, which is:
\[ y - y_1 = m(x - x_1) \]
Substituting \( A(1, 2) \) and the slope \( m_{AC} = 2 \):
\[ y - 2 = 2(x - 1) \]
Expanding this:
\[ y - 2 = 2x - 2 \]
\[ y = 2x \]
So, the equation of diagonal AC is \( y = 2x \).

### Step 5: Find the intersection of diagonals BD and AC
To find the coordinates of the center of the square, we need to find the intersection point of the two diagonals. We will solve the equations:
1. \( x + 2y = 10 \) (equation of diagonal BD)
2. \( y = 2x \) (equation of diagonal AC)

Substituting \( y = 2x \) into the first equation:
\[ x + 2(2x) = 10 \]
\[ x + 4x = 10 \]
\[ 5x = 10 \]
\[ x = 2 \]

Now, substituting \( x = 2 \) back into the equation of diagonal AC to find \( y \):
\[ y = 2(2) = 4 \]
Thus, the coordinates of the center of the square are \( (2, 4) \).

### Final Answers
- The equation of the other diagonal (AC) is: \( y = 2x \)
- The coordinates of the center of the square are: \( (2, 4) \)

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