Answer :
To find the area of the triangle formed by the line [tex]\(\frac{x}{m} + \frac{y}{n} = 1\)[/tex] with the coordinate axes, we can follow these steps:
1. Identify Intercepts:
- When [tex]\( y = 0 \)[/tex], the line intersects the x-axis at [tex]\( x = m \)[/tex].
- When [tex]\( x = 0 \)[/tex], the line intersects the y-axis at [tex]\( y = n \)[/tex].
2. Determine Base and Height:
- The intercept on the x-axis, [tex]\( (m, 0) \)[/tex], gives the length of the base of the triangle, which is [tex]\( m \)[/tex].
- The intercept on the y-axis, [tex]\( (0, n) \)[/tex], gives the height of the triangle, which is [tex]\( n \)[/tex].
3. Calculate the Area:
- The formula for the area of a triangle is [tex]\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)[/tex].
- Plugging in the base [tex]\( m \)[/tex] and the height [tex]\( n \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times m \times n = \frac{mn}{2} \][/tex]
4. Match the Result with Given Options:
- The calculated area [tex]\(\frac{mn}{2}\)[/tex] corresponds to option (c).
Thus, the area of the triangle formed by the line [tex]\(\frac{x}{m} + \frac{y}{n} = 1\)[/tex] with the coordinate axes is:
[tex]\[ \boxed{\frac{mn}{2}} \][/tex]
1. Identify Intercepts:
- When [tex]\( y = 0 \)[/tex], the line intersects the x-axis at [tex]\( x = m \)[/tex].
- When [tex]\( x = 0 \)[/tex], the line intersects the y-axis at [tex]\( y = n \)[/tex].
2. Determine Base and Height:
- The intercept on the x-axis, [tex]\( (m, 0) \)[/tex], gives the length of the base of the triangle, which is [tex]\( m \)[/tex].
- The intercept on the y-axis, [tex]\( (0, n) \)[/tex], gives the height of the triangle, which is [tex]\( n \)[/tex].
3. Calculate the Area:
- The formula for the area of a triangle is [tex]\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)[/tex].
- Plugging in the base [tex]\( m \)[/tex] and the height [tex]\( n \)[/tex]:
[tex]\[ \text{Area} = \frac{1}{2} \times m \times n = \frac{mn}{2} \][/tex]
4. Match the Result with Given Options:
- The calculated area [tex]\(\frac{mn}{2}\)[/tex] corresponds to option (c).
Thus, the area of the triangle formed by the line [tex]\(\frac{x}{m} + \frac{y}{n} = 1\)[/tex] with the coordinate axes is:
[tex]\[ \boxed{\frac{mn}{2}} \][/tex]
