Given Δ𝑃𝑄𝑅 with 𝑃(4;3), 𝑄(3;−2) and R(-2;0). The equation of line 𝑃𝑄 is 𝑦=5𝑥+17 and D is the 𝑥 intercept of line 𝑃𝑄. 𝑀(𝑥;𝑦) is midpoint of PR. Determine the angle of inclination of line 𝑃𝑄.

To determine the angle of inclination of the line \( PQ \), we need to find the angle that line \( PQ \) makes with the positive x-axis.

### Step-by-Step Solution:

1. **Equation of the Line**:  
   The equation of line \( PQ \) is given as:  
   \[
   y = 5x + 17
   \]
   Here, the slope \( m \) of the line is 5 (since the equation is in the slope-intercept form \( y = mx + c \)).

2. **Formula for Angle of Inclination**:  
   The angle of inclination \( \theta \) of a line with the positive x-axis is given by:  
   \[
   \theta = \arctan(m)
   \]
   where \( m \) is the slope of the line.

3. **Substitute the Slope \( m = 5 \)**:  
   We can calculate the angle of inclination using the arctangent function:  
   \[
   \theta = \arctan(5)
   \]

Let's compute the angle of inclination \( \theta \).The angle of inclination of line \( PQ \) with the positive x-axis is approximately \( 78.69^\circ \).

Would you like more details or have any questions about this calculation? Here are some related questions you might find interesting:

1. How do you find the slope of a line given two points on the line?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you convert an angle from radians to degrees?
4. How can you determine the x-intercept and y-intercept of a line?
5. What is the significance of the angle of inclination in geometry and trigonometry?

**Tip:** The arctangent function gives angles in radians by default, so remember to convert to degrees if needed!

Discover how to calculate the angle of inclination of line PQ with a step-by-step solution. The line equation \( y = 5x + 17 \) and points P(4, 3), Q(3, -2), and R(-2, 0) are used to determine the angle. Suitable for high school students studying geometry and trigonometry.

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