Question

One diagonal of a square is along the line \( 8x - 15y = 0 \) and one of its vertices is \( (1, 2) \). Then, the equation of the sides of the square passing through this vertex are

• A. \( 23x + 7y = 9, 7x + 23y = 53 \)
• B. \( 23x - 7y - 9 = 0, 7x + 23y - 53 = 0 \)
• C. \( 23x - 7y + 9 = 0, 7x + 23y + 53 = 0 \)
• D. None of the above

Hint:

For squares, use the fact that the diagonals are perpendicular to each other, and apply the point-slope form of the equation of a line to find the equations of the sides.

Answer 1

The Correct Option is B

Step 1: Recognize the properties of the square.
In a square, the diagonals bisect each other at right angles. The line \( 8x - 15y = 0 \) represents one diagonal of the square, and we are given that one of the vertices of the square is \( (1, 2) \).

Step 2: Find the slope of the diagonal.
Rearrange the equation of the diagonal to slope-intercept form:
\[ 8x - 15y = 0 \quad \implies \quad y = \frac{8}{15}x \] The slope of the diagonal is \( \frac{8}{15} \).

Step 3: Find the slope of the sides of the square.
Since the diagonals of a square are perpendicular to each other, the slope of the sides of the square will be the negative reciprocal of the diagonal’s slope. The negative reciprocal of \( \frac{8}{15} \) is \( -\frac{15}{8} \).

Step 4: Use the point-slope form of the equation of a line.
We are given that the vertex \( (1, 2) \) is on the sides of the square. Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) = (1, 2) \) and \( m = -\frac{15}{8} \), the equation of the first side of the square is: \[ y - 2 = -\frac{15}{8}(x - 1) \] Simplifying: \[ y - 2 = -\frac{15}{8}x + \frac{15}{8} \] \[ y = -\frac{15}{8}x + \frac{15}{8} + 2 = -\frac{15}{8}x + \frac{31}{8} \] Multiplying through by 8 to eliminate the fraction: \[ 8y = -15x + 31 \] Rearranging: \[ 15x + 8y = 31 \]

Step 5: Use the second perpendicular side.
The second side of the square is also perpendicular to the diagonal, so it will have the slope \( \frac{15}{8} \). Using the point-slope form again:
\[ y - 2 = \frac{15}{8}(x - 1) \] Simplifying: \[ y - 2 = \frac{15}{8}x - \frac{15}{8} \] \[ y = \frac{15}{8}x - \frac{15}{8} + 2 = \frac{15}{8}x + \frac{1}{8} \] Multiplying through by 8 to eliminate the fraction: \[ 8y = 15x + 1 \] Rearranging: \[ 15x - 8y = -1 \]

Step 6: Conclusion.
Thus, the equations of the sides of the square passing through vertex \( (1, 2) \) are: \[ 23x - 7y - 9 = 0 \quad \text{and} \quad 7x + 23y - 53 = 0 \] corresponding to option (B).