Understanding the Problem: Finding Triangle Area from Lines

The question asks us to calculate the area of the triangle formed by the intersection of three given lines: \(y = 3x\), \(y = 6x\), and \(y = 9\). To find the area of the triangle, we first need to determine the coordinates of its vertices. The vertices of the triangle are the points where these lines intersect each other.

Finding the Vertices of the Triangle

We need to find the intersection points for each pair of lines:

1. Intersection of Line \(y = 3x\) and Line \(y = 9\)

To find the intersection, we set the expressions for \(y\) equal to each other:

\(3x = 9\)

Solving for \(x\):

\(x = \frac{9}{3}\)

\(x = 3\)

The y-coordinate is given as 9. So, the first vertex is \((3, 9)\).

2. Intersection of Line \(y = 6x\) and Line \(y = 9\)

Again, we set the expressions for \(y\) equal:

\(6x = 9\)

Solving for \(x\):

\(x = \frac{9}{6}\)

\(x = \frac{3}{2}\)

The y-coordinate is 9. So, the second vertex is \((\frac{3}{2}, 9)\).

3. Intersection of Line \(y = 3x\) and Line \(y = 6x\)

Set the expressions for \(y\) equal:

\(3x = 6x\)

Rearrange the equation to solve for \(x\):

\(6x - 3x = 0\)

\(3x = 0\)

\(x = 0\)

Substitute \(x = 0\) into either equation (e.g., \(y = 3x\)):

\(y = 3(0)\)

\(y = 0\)

So, the third vertex is \((0, 0)\).

The three vertices of the triangle formed by the lines \(y = 3x\), \(y = 6x\), and \(y = 9\) are \((3, 9)\), \((\frac{3}{2}, 9)\), and \((0, 0)\).

Calculating the Area of the Triangle

We can calculate the area of the triangle using the coordinates of its vertices. Notice that two vertices, \((3, 9)\) and \((\frac{3}{2}, 9)\), share the same y-coordinate (9). This means the segment connecting these two points is a horizontal line segment, which can be considered the base of the triangle. The line segment lies on the line \(y = 9\).

Calculating the Base Length

The length of the base is the distance between the x-coordinates of the two points \((3, 9)\) and \((\frac{3}{2}, 9)\):

Base length \(b = |3 - \frac{3}{2}|\)

\(b = |\frac{6}{2} - \frac{3}{2}|\)

\(b = |\frac{3}{2}|\)

\(b = \frac{3}{2}\)

Calculating the Height

The height of the triangle is the perpendicular distance from the third vertex \((0, 0)\) to the line containing the base, which is the line \(y = 9\). The distance from a point \((x_0, y_0)\) to a horizontal line \(y = c\) is \(|y_0 - c|\).

Height \(h = |0 - 9|\)

\(h = |-9|\)

\(h = 9\)

Using the Area Formula

The area of a triangle is given by the formula:

Area \( = \frac{1}{2} \times \text{base} \times \text{height}\)

Substitute the calculated values for base and height:

Area \( = \frac{1}{2} \times \frac{3}{2} \times 9\)

Area \( = \frac{1 \times 3 \times 9}{2 \times 2}\)

Area \( = \frac{27}{4}\)

The area of the triangle formed by the lines is \(\frac{27}{4}\) square units.

Revision Table: Key Concepts

Additional Information: Lines, Vertices, and Area Calculation Methods

Understanding how to find the area of a triangle formed by intersecting lines is a fundamental concept in coordinate geometry. The steps involve finding the intersection points (vertices) and then using a suitable method to calculate the area.

Besides the base-and-height method used above, another common way to find the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is using the determinant formula:

Area \( = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\)

Let's quickly verify using this formula with our vertices \((3, 9)\), \((\frac{3}{2}, 9)\), and \((0, 0)\):

Let \((x_1, y_1) = (3, 9)\), \((x_2, y_2) = (\frac{3}{2}, 9)\), \((x_3, y_3) = (0, 0)\).

Area \( = \frac{1}{2} |3(9 - 0) + \frac{3}{2}(0 - 9) + 0(9 - 9)|\)

Area \( = \frac{1}{2} |3(9) + \frac{3}{2}(-9) + 0(0)|\)

Area \( = \frac{1}{2} |27 - \frac{27}{2}|\)

Area \( = \frac{1}{2} |\frac{54}{2} - \frac{27}{2}|\)

Area \( = \frac{1}{2} |\frac{27}{2}|\)

Area \( = \frac{27}{4}\)

Both methods confirm the area is \(\frac{27}{4}\) square units.

The lines \(y = 3x\) and \(y = 6x\) are lines passing through the origin \((0,0)\) with different slopes. The line \(y = 9\) is a horizontal line. Visualizing these lines can help understand the shape and position of the triangle formed.