Find the area of a triangle bounded by the y

axis, the line f(x)=-2/9x+5

, and the line perpendicular to f(x)

that passes through the origin.

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.

Learn how to find the area of a triangle bounded by the y-axis, the line f(x) = -2/9x + 5, and the perpendicular line passing through the origin. This problem involves geometry and coordinate geometry, focusing on calculating the area using the vertices and appropriate formulas. Suitable for high school students.

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