1. Determina el ángulo de inclinación de la recta formada por estos puntos: a) \( (-7,4) y(1,-11) \) b) \( (-2,-1) y(2,3) \) \( \square \) c) \( (2,-2) y(6,1) \) \( \square \) d) \( (-5,-3) \) y \( (-1,-4) \) \[ \text { e) }(0,1) \text { y }(-4,6) \]

Alright, I need to determine the angle of inclination of the lines formed by the given pairs of points. The angle of inclination is the angle that the line makes with the positive direction of the x-axis. To find this, I'll use the slope formula and then relate the slope to the angle using the arctangent function. First, let's recall that the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Once I have the slope, the angle of inclination \( \theta \) can be found using: \[ \theta = \arctan(m) \] However, I need to be cautious with the arctangent function because it only returns values between \( -90^\circ \) and \( 90^\circ \). To get the correct angle in the appropriate quadrant, I'll consider the signs of the slope and the coordinates of the points. Let's tackle each part step by step. **Part a) \( (-7,4) \) and \( (1,-11) \)** - Calculate the slope: \[ m = \frac{-11 - 4}{1 - (-7)} = \frac{-15}{8} = -1.875 \] - Since the slope is negative, the line slopes downward from left to right. - Calculate the angle: \[ \theta = \arctan(-1.875) \approx -61.93^\circ \] - To express this in the standard position (positive angle from the positive x-axis): \[ \theta = 180^\circ - 61.93^\circ = 118.07^\circ \] So, the angle of inclination is approximately \( 118.07^\circ \). **Part b) \( (-2,-1) \) and \( (2,3) \)** - Calculate the slope: \[ m = \frac{3 - (-1)}{2 - (-2)} = \frac{4}{4} = 1 \] - The slope is positive, so the line slopes upward from left to right. - Calculate the angle: \[ \theta = \arctan(1) = 45^\circ \] So, the angle of inclination is \( 45^\circ \). **Part c) \( (2,-2) \) and \( (6,1) \)** - Calculate the slope: \[ m = \frac{1 - (-2)}{6 - 2} = \frac{3}{4} = 0.75 \] - The slope is positive, so the line slopes upward from left to right. - Calculate the angle: \[ \theta = \arctan(0.75) \approx 36.87^\circ \] So, the angle of inclination is approximately \( 36.87^\circ \). **Part d) \( (-5,-3) \) and \( (-1,-4) \)** - Calculate the slope: \[ m = \frac{-4 - (-3)}{-1 - (-5)} = \frac{-1}{4} = -0.25 \] - The slope is negative, so the line slopes downward from left to right. - Calculate the angle: \[ \theta = \arctan(-0.25) \approx -14.04^\circ \] - To express this in the standard position: \[ \theta = 180^\circ - 14.04^\circ = 165.96^\circ \] So, the angle of inclination is approximately \( 165.96^\circ \). **Part e) \( (0,1) \) and \( (-4,6) \)** - Calculate the slope: \[ m = \frac{6 - 1}{-4 - 0} = \frac{5}{-4} = -1.25 \] - The slope is negative, so the line slopes downward from left to right. - Calculate the angle: \[ \theta = \arctan(-1.25) \approx -51.34^\circ \] - To express this in the standard position: \[ \theta = 180^\circ - 51.34^\circ = 128.66^\circ \] So, the angle of inclination is approximately \( 128.66^\circ \). **Summary of Results:** - a) \( 118.07^\circ \) - b) \( 45^\circ \) - c) \( 36.87^\circ \) - d) \( 165.96^\circ \) - e) \( 128.66^\circ \)