Vertices of a Triangle – Graphical Method
Video Explanation
Question
Determine graphically the vertices of the triangle, the equations of whose sides are:
\[ 2y – x = 8, \quad 5y – x = 14, \quad y – 2x = 1 \]
Solution
Step 1: Write the Given Equations in Convenient Form
Equation (1):
\[ 2y – x = 8 \Rightarrow y = \frac{x + 8}{2} \]
Equation (2):
\[ 5y – x = 14 \Rightarrow y = \frac{x + 14}{5} \]
Equation (3):
\[ y – 2x = 1 \Rightarrow y = 2x + 1 \]
Step 2: Find Points for Each Line (for Graph)
For \(2y – x = 8\)
| x | y |
|---|---|
| 0 | 4 |
| 2 | 5 |
For \(5y – x = 14\)
| x | y |
|---|---|
| 4 | 6 |
| -1 | \(\frac{13}{5}\) |
For \(y – 2x = 1\)
| x | y |
|---|---|
| 0 | 1 |
| 1 | 3 |
Step 3: Find the Vertices (Points of Intersection)
Vertex A: Intersection of \(2y – x = 8\) and \(5y – x = 14\)
Subtracting:
\[ (5y – x) – (2y – x) = 14 – 8 \Rightarrow 3y = 6 \Rightarrow y = 2 \]
Substitute in \(2y – x = 8\):
\[ 2(2) – x = 8 \Rightarrow x = -4 \]
\[ A(-4,\,2) \]
Vertex B: Intersection of \(5y – x = 14\) and \(y – 2x = 1\)
From \(y = 2x + 1\), substitute in \(5y – x = 14\):
\[ 5(2x + 1) – x = 14 \Rightarrow 10x + 5 – x = 14 \Rightarrow 9x = 9 \Rightarrow x = 1 \]
\[ y = 2(1) + 1 = 3 \]
\[ B(1,\,3) \]
Vertex C: Intersection of \(2y – x = 8\) and \(y – 2x = 1\)
From \(y = 2x + 1\), substitute in \(2y – x = 8\):
\[ 2(2x + 1) – x = 8 \Rightarrow 4x + 2 – x = 8 \Rightarrow 3x = 6 \Rightarrow x = 2 \]
\[ y = 2(2) + 1 = 5 \]
\[ C(2,\,5) \]
Conclusion
The vertices of the triangle are:
\[ \boxed{A(-4,\,2), \; B(1,\,3), \; C(2,\,5)} \]
These points are obtained as the intersection points of the given lines and can be verified graphically.