Vertices of a Triangle by Graphical Method
Video Explanation
Question
Determine graphically the vertices of the triangle, the equations of whose sides are:
\[ y = x,\quad y = 0,\quad 3x + 3y = 10 \]
Solution
Step 1: Write the Given Equations
Line (1): \(\; y = x\)
Line (2): \(\; y = 0\)
Line (3): \(\; 3x + 3y = 10 \Rightarrow y = \frac{10}{3} – x\)
Step 2: Prepare Points for Graphical Representation
For \(y = x\)
| x | y |
|---|---|
| 0 | 0 |
| 2 | 2 |
For \(y = 0\)
| x | y |
|---|---|
| 0 | 0 |
| 3 | 0 |
For \(3x + 3y = 10\)
| x | y |
|---|---|
| 0 | \(\frac{10}{3}\) |
| \(\frac{10}{3}\) | 0 |
Step 3: Find the Vertices (Points of Intersection)
Vertex A: Intersection of \(y = x\) and \(y = 0\)
\[ y = 0 \Rightarrow x = 0 \]
\[ A(0,\,0) \]
Vertex B: Intersection of \(y = 0\) and \(3x + 3y = 10\)
\[ 3x = 10 \Rightarrow x = \frac{10}{3} \]
\[ B\left(\frac{10}{3},\,0\right) \]
Vertex C: Intersection of \(y = x\) and \(3x + 3y = 10\)
\[ 3x + 3x = 10 \Rightarrow 6x = 10 \Rightarrow x = \frac{5}{3} \]
\[ y = x = \frac{5}{3} \]
\[ C\left(\frac{5}{3},\,\frac{5}{3}\right) \]
Conclusion
The vertices of the triangle are:
\[ \boxed{A(0,\,0),\; B\left(\frac{10}{3},\,0\right),\; C\left(\frac{5}{3},\,\frac{5}{3}\right)} \]
These vertices are obtained as the points of intersection of the given lines and can be verified graphically.