Find the area of `Delta` formed by three lines `8x+15y=120`, x axis and y axis

Answer

To find the area of the triangle formed by the lines \(8x + 15y = 120\), the x-axis, and the y-axis, we can follow these steps:

### Step 1: Find the x-intercept and y-intercept of the line
To find the intercepts, we can rewrite the equation of the line in intercept form.

1. Set \(y = 0\) to find the x-intercept:
\[ 8x + 15(0) = 120 \implies 8x = 120 \implies x = \frac{120}{8} = 15 \]
Thus, the x-intercept is \( (15, 0) \).

2. Set \(x = 0\) to find the y-intercept:
\[ 8(0) + 15y = 120 \implies 15y = 120 \implies y = \frac{120}{15} = 8 \]
Thus, the y-intercept is \( (0, 8) \).

### Step 2: Identify the vertices of the triangle
The vertices of the triangle formed by the line and the axes are:
- A (0, 0) - the origin
- B (15, 0) - the x-intercept
- C (0, 8) - the y-intercept

### Step 3: Calculate the area of the triangle
The area \(A\) of a triangle formed by the x-axis and y-axis can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base is the length along the x-axis (BC) which is 15 units, and the height is the length along the y-axis (AC) which is 8 units.

Substituting these values into the formula:
\[ A = \frac{1}{2} \times 15 \times 8 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 120 = 60 \text{ square units} \]

### Final Answer
The area of the triangle formed by the lines \(8x + 15y = 120\), the x-axis, and the y-axis is \(60\) square units.

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