Graphical Determination of the Vertices of a Triangle

Video Explanation

Question

Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are:

\[ y = x \]

\[ 3y = x \]

\[ x + y = 8 \]

Solution

Step 1: Write the Equations in Convenient Form

Equation (1): \(y = x\)

Equation (2):

\[ 3y = x \Rightarrow y = \frac{x}{3} \]

Equation (3):

\[ x + y = 8 \Rightarrow y = 8 – x \]

Step 2: Find the Points of Intersection

Intersection of \(y = x\) and \(y = \frac{x}{3}\)

\[ x = \frac{x}{3} \Rightarrow 3x = x \Rightarrow 2x = 0 \Rightarrow x = 0 \]

\[ y = 0 \]

Intersection point = \((0, 0)\)

Intersection of \(y = x\) and \(y = 8 – x\)

\[ x = 8 – x \Rightarrow 2x = 8 \Rightarrow x = 4 \]

\[ y = 4 \]

Intersection point = \((4, 4)\)

Intersection of \(y = \frac{x}{3}\) and \(y = 8 – x\)

\[ \frac{x}{3} = 8 – x \Rightarrow x + 3x = 24 \Rightarrow 4x = 24 \Rightarrow x = 6 \]

\[ y = \frac{6}{3} = 2 \]

Intersection point = \((6, 2)\)

Step 3: Graphical Representation

Plot the three straight lines \(y = x\), \(y = \frac{x}{3}\) and \(y = 8 – x\) on the same Cartesian plane. The three lines intersect pairwise to form a triangle.

Answer

The coordinates of the vertices of the triangle are:

• \((0, 0)\)
• \((4, 4)\)
• \((6, 2)\)

Conclusion

Thus, the triangle formed by the given equations has vertices at \((0, 0)\), \((4, 4)\) and \((6, 2)\).