Answer:
Given that the equation of diagonal AC of square ABCD is \(3x - 4y + 10 = 0\) and the coordinates of vertex B are \((4, -5)\), we need to find the equation of the other diagonal, BD.
Here are the steps to find the equation of diagonal BD:
1. **Find the midpoint of diagonal AC**:
- The diagonals of a square bisect each other at right angles and intersect at the square's center.
- Let's denote the center (intersection point of the diagonals) as \(M\).
2. **Slope of diagonal AC**:
- Rewrite the equation \(3x - 4y + 10 = 0\) in slope-intercept form: \(y = \frac{3}{4}x + \frac{10}{4}\).
3. **Slope of diagonal BD**:
- Diagonals in a square are perpendicular. Thus, if the slope of AC is \(\frac{3}{4}\), the slope of BD will be the negative reciprocal, which is \(-\frac{4}{3}\).
4. **Equation of diagonal BD**:
- Using the slope-point form of the equation of a line with the slope \(-\frac{4}{3}\) passing through \(B(4, -5)\):
y - (-5) = -\frac{4}{3}(x - 4)
Simplify this equation to get the equation of BD:
y + 5 = -\frac{4}{3}(x - 4)
y + 5 = -\frac{4}{3}x + \frac{16}{3}
3(y + 5) = -4(x - 4)
3y + 15 = -4x + 16
4x + 3y = 1
Thus, the equation of diagonal BD is \(4x + 3y = 1\).
Step-by-step explanation:
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