The sides of a triangle are given by the equation `y-2=0,y+1=3(x-2) and x+2y=0` Find, graphically:
The area of a triangle

Answer

To find the area of the triangle formed by the given equations graphically, we will follow these steps:

### Step 1: Identify the equations of the sides of the triangle
The equations given are:
1. \( y - 2 = 0 \)
2. \( y + 1 = 3(x - 2) \)
3. \( x + 2y = 0 \)

### Step 2: Convert the equations to slope-intercept form
1. For the first equation \( y - 2 = 0 \):
\[ y = 2 \]
This is a horizontal line at \( y = 2 \).

2. For the second equation \( y + 1 = 3(x - 2) \):
\[ y + 1 = 3x - 6 \implies y = 3x - 7 \]

3. For the third equation \( x + 2y = 0 \):
\[ 2y = -x \implies y = -\frac{1}{2}x \]

### Step 3: Find the points of intersection of the lines
To find the vertices of the triangle, we need to find the intersection points of the lines.

1. **Intersection of \( y = 2 \) and \( y = 3x - 7 \)**:
\[ 2 = 3x - 7 \implies 3x = 9 \implies x = 3 \]
So, the point is \( (3, 2) \).

2. **Intersection of \( y = 2 \) and \( y = -\frac{1}{2}x \)**:
\[ 2 = -\frac{1}{2}x \implies x = -4 \]
So, the point is \( (-4, 2) \).

3. **Intersection of \( y = 3x - 7 \) and \( y = -\frac{1}{2}x \)**:
\[ 3x - 7 = -\frac{1}{2}x \implies 3.5x = 7 \implies x = 2 \]
Substituting \( x = 2 \) into \( y = 3x - 7 \):
\[ y = 3(2) - 7 = 6 - 7 = -1 \]
So, the point is \( (2, -1) \).

### Step 4: Identify the vertices of the triangle
The vertices of the triangle are:
- \( A(3, 2) \)
- \( B(-4, 2) \)
- \( C(2, -1) \)

### Step 5: Calculate the area of the triangle
The area \( A \) of a triangle given vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates:
\[ A = \frac{1}{2} \left| 3(2 - (-1)) + (-4)(-1 - 2) + 2(2 - 2) \right| \]
Calculating each term:
\[ = \frac{1}{2} \left| 3(3) + (-4)(-3) + 2(0) \right| \]
\[ = \frac{1}{2} \left| 9 + 12 + 0 \right| = \frac{1}{2} \left| 21 \right| = \frac{21}{2} \]
Thus, the area of the triangle is \( 10.5 \) square units.