03_Ex 4.3

Exercise Questions

Questions	Links
Q1. Find the slope and angle of…	4.3 Q-1
Q2. In the triangle $A(8,6), C(-2,-6)$ …	4.3 Q-2
Q3. By means of slopes, show that…	4.3 Q-3
Q4. Find $K$ so the line joining…	4.3 Q-4
Q5. Using Slope, show that the $D$ with vertices…	4.3 Q-5
Q6. The points $A(7,1), B(-2,2)$ and…	4.3 Q-6
Q7. The point $A(-1,2), B(3,-1)$ and…	4.3 Q-7
Q8. Two Pairs of points are given. Find…	4.3 Q-8
Q9. Find an equation of…	4.3 Q-9
Q10. Find an equation of the line…	4.3 Q-10
Q11. Find an equation of the perpendicular bisector…	4.3 Q-11
Q12. Find an equation of the sides, altitudes and…	4.3 Q-12
Q13. Find an equation of the line through $(-4,-6)$ …	4.3 Q-13
Q14. Find an equation of line through $(11,-5)$ …	4.3 Q-14
Q15. The point $A(-1,2), B(6,3)$ and $C(2,-4)$ are…	4.3 Q-15
Q16. A milk man can sell 500 liters of milk…	4.3 Q-16
Q17. The Population of Pakistan to the nearest…	4.3 Q-17
Q18. A house was purchase for Rs. 1 million in 1980…	4.3 Q-18
Q19. Plot the Celsius (C) and Fahrenheit (F) temperature…	4.3 Q-19
Q20. The average entry test score of engineering…	4.3 Q-20
Q21. Convert each of the following equation…	4.3 Q-21
Q22. Check whether the two lines are:	4.3 Q-22
Q23. Find the distance between the given parallel…	4.3 Q-23
Q24. Find an equation of line through…	4.3 Q-24
Q25. Find an equation of the line through $(5,-8)$ …	4.3 Q-25
Q26. Find an equation of two parallel line…	4.3 Q-26
Q27. One vertex of a parallelogram is…	4.3 Q-27
Q28. Find whether the given point lies above…	4.3 Q-28
Q29. Check whether the given points…	4.3 Q-29
Q30. Find the distance from the point $P(6,-1)$ …	4.3 Q-30
Q31. Find the area of the triangular region	4.3 Q-31
Q32. The coordinates of three points are…	4.3 Q-32

Overview

This exercise focuses on understanding the concepts of slopes and angles of inclination. These concepts are vital in analyzing the properties of lines and their relations in geometry. The ability to calculate slopes and angles allows us to determine if lines are parallel, perpendicular, or collinear, which are essential in various fields, including engineering, physics, and computer graphics.

Key Concepts

Inclination: The angle formed between a line and the positive direction of the x-axis.
- Range of $\theta$ :
  $0 \leq \theta \leq 180^\circ$
Slope ( $m$ ): The steepness of a line, calculated as the ratio of rise to run.
- Formula:
  $m = \frac{y}{x} = \tan \theta$
- Two-Point Slope Formula: Given two points $(x_1, y_1)$ and $(x_2, y_2)$ , the slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Collinearity: Points are collinear if the slopes between consecutive points are equal.
Parallel Lines: Lines with equal slopes.
Perpendicular Lines: Lines where the product of their slopes equals -1. $m_1 \cdot m_2 = -1$

Forms

Slope-intercept form: This is of the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept.
Intercept form: This is expressed as $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ and $b$ are the x- and y-intercepts, respectively.
Normal form: This form is $\frac{Ax}{\sqrt{A^2 + B^2}} + \frac{By}{\sqrt{A^2 + B^2}} = \frac{C}{\sqrt{A^2 + B^2}}$ , which expresses the line in terms of its normal vector.

Important Formulas

Slope of a Line:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Inclination Angle:
$\theta = \tan^{-1}(m)$
Collinearity:
$m_{AB} = m_{BC}$
Parallel Lines:
$m_1 = m_2$
Perpendicular Lines:
$m_1 \cdot m_2 = -1$
Slope of Altitude:
$\text{slope} = \frac{-1}{m_{\text{line}}}$

Tips and Tricks

For undefined slope: If the denominator of the slope formula is zero (i.e., vertical line), the slope is considered undefined.
For zero slope: If the numerator of the slope formula is zero (i.e., horizontal line), the slope is zero.
Collinearity check: To determine if points are collinear, check if the slopes between consecutive points are equal.
For perpendicular lines: If the product of the slopes is -1, the lines are perpendicular.

Summary

In this exercise, we explored the calculation of slopes and angles of inclination for various line configurations. We also covered important geometric concepts such as collinearity, parallelism, and perpendicularity. By using these principles, one can determine relationships between lines, which is crucial in solving geometry problems involving linear equations.

Reference

By Sir Shazad Sair:

By Great Science Academy

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