Triangle Area Calculation for Line $2x - 4y - 7 = 0$

To find the area of the triangle formed by the line $2x - 4y - 7 = 0$ with the coordinate axes, we first need to determine the points where the line intersects the x-axis and the y-axis. These points represent the vertices of the triangle along with the origin.

Finding Intercepts

• X-intercept: Set $y=0$ in the line equation.

  $2x - 4(0) - 7 = 0$

  $2x - 7 = 0$

  $2x = 7$

  $x = \frac{7}{2}$

  The x-intercept is at the point $(\frac{7}{2}, 0)$. The length of the base along the x-axis is $|\frac{7}{2}| = \frac{7}{2}$.
• Y-intercept: Set $x=0$ in the line equation.

  $2(0) - 4y - 7 = 0$

  $-4y - 7 = 0$

  $-4y = 7$

  $y = -\frac{7}{4}$

  The y-intercept is at the point $(0, -\frac{7}{4})$. The length of the height along the y-axis is $|-\frac{7}{4}| = \frac{7}{4}$.

Calculating Triangle Area

The area of the triangle formed by a line and the coordinate axes is given by the formula:

Area $= \frac{1}{2} \times |\text{x-intercept}| \times |\text{y-intercept}|$

Substitute the calculated intercept values:

Area $= \frac{1}{2} \times \frac{7}{2} \times \frac{7}{4}$

Area $= \frac{1 \times 7 \times 7}{2 \times 2 \times 4}$

Area $= \frac{49}{16}$

Therefore, the area of the triangle is $\frac{49}{16} \text{ unit}^2$.