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:

\[ y = x \]

\[ y = 2x \]

\[ y + x = 6 \]

Solution

Step 1: Write the Equations in Convenient Form

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

Equation (2): \(y = 2x\)

Equation (3):

\[ y + x = 6 \Rightarrow y = 6 – x \]

Step 2: Find Points of Intersection

Intersection of \(y = x\) and \(y = 2x\)

\[ x = 2x \Rightarrow x = 0 \]

\[ y = 0 \]

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

Intersection of \(y = x\) and \(y = 6 – x\)

\[ x = 6 – x \Rightarrow 2x = 6 \Rightarrow x = 3 \]

\[ y = 3 \]

Intersection point = \((3, 3)\)

Intersection of \(y = 2x\) and \(y = 6 – x\)

\[ 2x = 6 – x \Rightarrow 3x = 6 \Rightarrow x = 2 \]

\[ y = 4 \]

Intersection point = \((2, 4)\)

Step 3: Graphical Representation

Plot the three straight lines \(y = x\), \(y = 2x\) and \(y = 6 – x\) on the same Cartesian plane. The three lines intersect pairwise to form a triangle.

Answer

The coordinates of the vertices of the triangle are:

• \((0, 0)\)
• \((3, 3)\)
• \((2, 4)\)

Conclusion

By drawing the graphs of the given equations, the triangle formed has vertices at \((0, 0)\), \((3, 3)\) and \((2, 4)\).