[B] ABCD is a rhombus in which \( A(5,3), B(6,-2), C(1, m) \) Find the value of \( m \) ?

To find the value of \( m \), we can use the property of a rhombus that its diagonals bisect each other at right angles. First, we find the midpoint of diagonal \( AC \) and diagonal \( BD \). Since \( D \) is unknown, we can find the midpoint of \( AC \) using points \( A(5,3) \) and \( C(1,m) \): \[ \text{Midpoint of } AC = \left( \frac{5 + 1}{2}, \frac{3 + m}{2} \right) = \left( \frac{6}{2}, \frac{3 + m}{2} \right) = (3, \frac{3 + m}{2}) \] Next, we find the midpoint of \( BD \). To find point \( D \), we first need the coordinates of \( B(6,-2) \). The coordinates of \( D \) can be denoted as \( D(x, y) \), and its midpoint with \( B \) is: \[ \text{Midpoint of } BD = \left( \frac{6 + x}{2}, \frac{-2 + y}{2} \right) \] Since these midpoints must be equal, we can set them equal to each other: \[ \frac{6 + x}{2} = 3 \quad \text{and} \quad \frac{-2 + y}{2} = \frac{3 + m}{2} \] From the first equation, solving for \( x \): \[ 6 + x = 6 \implies x = 0 \] From the second equation, solving for \( y \): \[ -2 + y = 3 + m \implies y = 5 + m \] Now we need to ensure that the diagonal \( AC \) and \( BD \) are perpendicular. The slope of \( AC \) is: \[ \text{slope of } AC = \frac{m - 3}{1 - 5} = \frac{m - 3}{-4} \] The slope of \( BD \) is: \[ \text{slope of } BD = \frac{(5 + m) - (-2)}{0 - 6} = \frac{m + 7}{-6} \] For \( AC \) and \( BD \) to be perpendicular, the product of their slopes must equal -1: \[ \frac{m - 3}{-4} \cdot \frac{m + 7}{-6} = -1 \] Simplifying this gives: \[ \frac{(m - 3)(m + 7)}{24} = -1 \] Thus, multiplying both sides by 24: \[ (m - 3)(m + 7) = -24 \] Expanding this, we have: \[ m^2 + 4m - 21 = 0 \] Now, factoring the quadratic: \[ (m - 3)(m + 7) = 0 \] Setting each factor to zero gives us: \[ m - 3 = 0 \Rightarrow m = 3 \] \[ m + 7 = 0 \Rightarrow m = -7 \] Now let's verify. The points \( C(1, 3) \) and \( C(1, -7) \) will sync with the properties of the rhombus. Finally, the value of \( m \) is \( 3 \) or \( -7 \). Since both are valid solutions from different vertices, we can report: So, the possible values of \( m \) are \( 3 \) or \( -7 \).