Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Coordinate geometry of straight lines
• Intersection points of linear equations
• Area of triangles using vertex coordinates
• Centroid of a triangle (intersection of medians)
• Circumcenter and perpendicular bisectors
• Distance from a point to a line
• Angle between two intersecting lines
• Applications of linear geometry to engineering and design problems

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Important Formulas

Below are the key formulas used in this exercise:

Area of Triangle (Coordinate Geometry): $\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$

Distance from Point to Line: $d=\frac{|a{x}_0+b{y}_0+c|}{\sqrt{a^2+b^2}}$ where $(x_0,y_0)$ is the point and $ax+by+c=0$ is the line.

Centroid of Triangle: $G=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$

Angle Between Two Lines: $\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$ where $m_1$ and $m_2$ are the slopes of the lines.

Slope of Line: $m=\frac{y_2-y_1}{x_2-x_1}$

Concurrency of Three Lines: Three lines $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2=0$, and $a_{3}x+b_{3}y+c_3=0$ are concurrent if the determinant of their coefficients is zero.

Homogeneous Equations and Joint Lines: A homogeneous equation of degree $n$ in $x$ and $y$ always represents $n$ straight lines passing through the origin.

For the second-degree equation $a{x}^2+2hxy+b{y}^2=0$:

The lines are real and distinct if $h^2-ab>0$.

Individual lines can be extracted by factoring or using the quadratic formula.

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Summary

This exercise applies coordinate geometry to solve practical problems involving triangular configurations and linear paths. Key strategies include using the determinant formula for triangular area, solving systems of linear equations to find intersection points (vertices), and calculating special points like the centroid (center of mass) and circumcenter (equidistant point). The problems emphasize converting real-world scenarios (flight paths, land boundaries, structural supports) into algebraic representations and using distance and angle formulas to verify geometric properties. Mastery of simultaneous equations and the point-to-line distance formula is essential for solving optimization and positioning problems.