01_Ex 4.1

Exercise Questions

Questions	Links
Q1. Describe the location in the plane…	4.1 Q-1
Q2. Find in each of the following…	4.1 Q-2
Q3. Which of the following points are at…	4.1 Q-3
Q4. Show that…	4.1 Q-4
Q5. The mid-Point of the sides of a triangle…	4.1 Q-5
Q6. Find $h$ such that the point $A(\sqrt{3},-1)$ …	4.1 Q-6
Q7. Find $h$ such that $A(-1, h)$ …	4.1 Q-7
Q8. The points $A(-5,-2)$ and $B(5,-4)$ are…	4.1 Q-8
Q9. Find $h$ such that the point $A(h, 1)$ ..	4.1 Q-9
Q10. A quadrilateral has the points $A(9,3)$ …	4.1 Q-10
Q11. Find $h$ such that the quadrilateral…	4.1 Q-11
Q12. If two vertices of an equilateral triangle are…	4.1 Q-12
Q13. Find the points trisecting…	4.1 Q-13
Q14. Find the point threes fifth of the way…	4.1 Q-14
Q15. Find the point on the join of $A(1,4)$ …	4.1 Q-15
Q16. Find the point which is equidistant…	4.1 Q-16
Q17. The point $(4,-2),(-2,4)$ and…	4.1 Q-17
Q18. Find the point that divide the line segment…	4.1 Q-18

Overview

This exercise introduces essential concepts in analytic geometry, focusing on understanding the relationships between points, lines, and shapes in a coordinate plane. By applying formulas such as distance, midpoint, and slope, students learn to analyze geometric properties, identify types of triangles and quadrilaterals, and solve problems involving their dimensions and orientations. These concepts are fundamental to advanced geometry and its applications in real-world scenarios.

Key Concepts

Distance Formula: Used to find the length of a line segment between two points.
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint Formula: Calculates the midpoint of a line segment joining two points.
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Slope of a Line: Measures the steepness or incline of a line.
$m = \frac{y_2 - y_1}{x_2 - x_1}$
- Parallel Lines: Equal slopes.
- Perpendicular Lines: Product of slopes equals $-1$ .
Pythagorean Theorem: Determines whether a triangle is right-angled.
$c^2 = a^2 + b^2$
Classification of Shapes:
- Right Triangle: Verified using distance or slope properties.
- Isosceles Triangle: Two sides are equal in length.
- Parallelogram: Opposite sides are equal, and diagonals bisect each other.
- Square: All sides are equal, and diagonals are equal and perpendicular.

Important Formulas

Distance Formula:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint Formula:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Slope of a Line:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Pythagorean Theorem:
$c^2 = a^2 + b^2$
Section Formula: To find the coordinates of a point dividing a line segment in the ratio $m:n$ :

x = \frac{n x_1 + m x_2}{m + n}, \quad y = \frac{n y_1 + m y_2}{m + n}

Tips and Tricks

Always label points and coordinates clearly to avoid confusion.
Use the distance formula to verify if sides of a shape are equal or to check right angles.
Midpoints can simplify problem-solving by identifying symmetry or dividing shapes into simpler sections.
Use slopes to quickly determine parallel or perpendicular lines without additional calculations.
For complex problems, sketch a rough diagram to visualize the points and relationships.

Summary

This exercise emphasizes the use of fundamental formulas in analytic geometry to solve problems involving points, lines, and polygons. Mastering these formulas allows students to classify shapes, verify geometric properties, and solve problems systematically. The distance and midpoint formulas, along with slope calculations, are tools for analyzing and understanding the geometric relationships in a coordinate plane.

Reference

By Sir Shahzad Sair:

By Great Science Academy: