To find the area of the triangle bounded by the y-axis, the line $f(x) = -\frac{2}{9}x + 5$, and the line perpendicular to $f(x)$ that passes through the origin, follow these steps:

1. Identify the slope of the perpendicular line:
   The given line has a slope of $-\frac{2}{9}$. The slope of a line perpendicular to this will be the negative reciprocal of $-\frac{2}{9}$, which is $\frac{9}{2}$.

2. Equation of the perpendicular line:
   Since this line passes through the origin $(0, 0)$, its equation is: $y = \frac{9}{2}x$

3. Find the intersection point of the two lines:
   Set the equations equal to find the intersection point. $-\frac{2}{9}x + 5 = \frac{9}{2}x$ Multiply through by 18 to clear the fractions: $-4x + 90 = 81x$ Combine like terms: $90 = 85x$ Solve for $x$: $x = \frac{90}{85} = \frac{18}{17}$ Substitute $x = \frac{18}{17}$ back into one of the line equations to find $y$: $y = \frac{9}{2} \left( \frac{18}{17} \right) = \frac{81}{17}$ So, the intersection point is $\left( \frac{18}{17}, \frac{81}{17} \right)$.

4. Find the vertices of the triangle:

   • One vertex is at the origin: $(0,0)$.
   • Another vertex is at the y-intercept of $f(x) = -\frac{2}{9}x + 5$, which is $(0, 5)$.
   • The third vertex is the intersection point: $\left( \frac{18}{17}, \frac{81}{17} \right)$.

5. Calculate the area of the triangle:
   Using the vertices $(0, 0)$, $(0, 5)$, and $\left( \frac{18}{17}, \frac{81}{17} \right)$, use the formula for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$ Substituting in our points: $\text{Area} = \frac{1}{2} \left| 0(5 - \frac{81}{17}) + 0(\frac{81}{17} - 0) + \frac{18}{17}(0 - 5) \right|$ Simplify inside the absolute value: $\text{Area} = \frac{1}{2} \left| \frac{18}{17}(-5) \right| = \frac{1}{2} \left| -\frac{90}{17} \right| = \frac{1}{2} \cdot \frac{90}{17} = \frac{45}{17}$

So, the area of the triangle is $\frac{45}{17}$ square units.

Would you like more details or have any questions?

Here are 8 related questions that expand on this information:

1. How do you find the equation of a line given a point and a slope?
2. What is the negative reciprocal of a slope, and why is it important for perpendicular lines?
3. How do you find the y-intercept of a line given its equation?
4. How do you solve for the intersection point of two lines?
5. What is the general formula for the area of a triangle given its vertices?
6. Can you explain the steps to convert a linear equation into slope-intercept form?
7. How would the process change if the triangle was bounded by different lines?
8. What are some real-world applications of finding the area of a triangle using coordinates?

Tip: Always double-check your calculations for intersection points and area, as small arithmetic errors can lead to incorrect results.