Find the area of triangle formed by straight lines `x + y - 4 = 0, x + 2y - 10 =0 and y =0`

Answer

To find the area of the triangle formed by the lines \( x + y - 4 = 0 \), \( x + 2y - 10 = 0 \), and \( y = 0 \), we will follow these steps:

### Step 1: Find the Intercepts of the Lines
1. **For the line \( x + y - 4 = 0 \)**:
- Set \( y = 0 \) to find the x-intercept:
\[ x + 0 - 4 = 0 \implies x = 4 \implies (4, 0) \]
- Set \( x = 0 \) to find the y-intercept:
\[ 0 + y - 4 = 0 \implies y = 4 \implies (0, 4) \]

2. **For the line \( x + 2y - 10 = 0 \)**:
- Set \( y = 0 \) to find the x-intercept:
\[ x + 0 - 10 = 0 \implies x = 10 \implies (10, 0) \]
- Set \( x = 0 \) to find the y-intercept:
\[ 0 + 2y - 10 = 0 \implies 2y = 10 \implies y = 5 \implies (0, 5) \]

### Step 2: Identify the Vertices of the Triangle
The vertices of the triangle formed by the lines are:
- Point A: \( (4, 0) \) (intersection of \( x + y - 4 = 0 \) and \( y = 0 \))
- Point B: \( (10, 0) \) (intersection of \( x + 2y - 10 = 0 \) and \( y = 0 \))
- Point C: Intersection of the lines \( x + y - 4 = 0 \) and \( x + 2y - 10 = 0 \)

### Step 3: Find the Intersection Point C
To find point C, we solve the equations:
1. \( x + y = 4 \) (from \( x + y - 4 = 0 \))
2. \( x + 2y = 10 \) (from \( x + 2y - 10 = 0 \))

Subtract the first equation from the second:
\[ (x + 2y) - (x + y) = 10 - 4 \implies y = 6 \]
Substituting \( y = 6 \) into \( x + y = 4 \):
\[ x + 6 = 4 \implies x = -2 \implies C(-2, 6) \]

### Step 4: Calculate the Area of the Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- The base is the distance between points A and B:
\[ \text{Base} = |10 - 4| = 6 \]
- The height is the y-coordinate of point C since it is perpendicular to the base:
\[ \text{Height} = 6 \]

Substituting the values into the area formula:
\[ A = \frac{1}{2} \times 6 \times 6 = \frac{36}{2} = 18 \]

### Final Answer
The area of the triangle formed by the lines is \( 18 \) square units.

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