11 has coordinates \( (4,1) \) and \( N \) has coordinates \( (-2,-7) \). (a) Find the length of \( M N \). (b) Find the gradient of \( M N \). (c) Find the equation of the perpendicular bisector of \( M N \). (c) (b) (b)

Let's solve the problem step by step. ### Given: - Point \( M \) has coordinates \( (4, 1) \). - Point \( N \) has coordinates \( (-2, -7) \). ### (a) Find the length of \( MN \). The length of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points \( M \) and \( N \): \[ d = \sqrt{((-2) - 4)^2 + ((-7) - 1)^2} \] Calculating the differences: \[ d = \sqrt{(-6)^2 + (-8)^2} \] Calculating the squares: \[ d = \sqrt{36 + 64} \] Adding the squares: \[ d = \sqrt{100} \] Taking the square root: \[ d = 10 \] ### (b) Find the gradient of \( MN \). The gradient (or slope) of a line segment between two points can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of points \( M \) and \( N \): \[ m = \frac{-7 - 1}{-2 - 4} \] Calculating the differences: \[ m = \frac{-8}{-6} \] Simplifying the fraction: \[ m = \frac{4}{3} \] ### (c) Find the equation of the perpendicular bisector of \( MN \). 1. **Find the midpoint of \( MN \)**: The midpoint \( P \) of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of points \( M \) and \( N \): \[ P = \left( \frac{4 + (-2)}{2}, \frac{1 + (-7)}{2} \right) \] Calculating the midpoint: \[ P = \left( \frac{2}{2}, \frac{-6}{2} \right) = (1, -3) \] 2. **Find the gradient of the perpendicular bisector**: The gradient of the perpendicular bisector is the negative reciprocal of the gradient of \( MN \): \[ m_{perpendicular} = -\frac{1}{m} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \] 3. **Use the point-slope form to find the equation**: The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Using the midpoint \( P(1, -3) \) and the gradient \( m_{perpendicular} = -\frac{3}{4} \): \[ y - (-3) = -\frac{3}{4}(x - 1) \] Simplifying: \[ y + 3 = -\frac{3}{4}x + \frac{3}{4} \] Subtracting 3 from both sides: \[ y = -\frac{3}{4}x + \frac{3}{4} - 3 \] Converting \( -3 \) to a fraction: \[ y = -\frac{3}{4}x + \frac{3}{4} - \frac{12}{4} \] Combining the fractions: \[ y = -\frac{3}{4}x - \frac{9}{4} \] ### Summary of Results: - (a) Length of \( MN \): \( 10 \) - (b) Gradient of \( MN \): \( \frac{4}{3} \) - (c) Equation of the perpendicular bisector: \( y = -\frac{3}{4}x - \frac{9}{4} \)