To solve question 19:

Problem Statement

The three sides of a triangle are represented by the lines:

1. $x = 1$
2. $y = 2$
3. $4x + 5y = 20$

We need to find the area of the triangle formed by these lines.

Solution

Step 1: Find the vertices of the triangle

The vertices are formed at the intersection points of these lines. Let us calculate them pair by pair:

1. Intersection of $x = 1$ and $y = 2$: $x = 1, \; y = 2 \implies (1, 2)$

2. Intersection of $x = 1$ and $4x + 5y = 20$: Substituting $x = 1$ into $4x + 5y = 20$: $4(1) + 5y = 20 \implies 5y = 16 \implies y = \frac{16}{5}$ Hence, the intersection is $\left(1, \frac{16}{5}\right)$.

3. Intersection of $y = 2$ and $4x + 5y = 20$: Substituting $y = 2$ into $4x + 5y = 20$: $4x + 5(2) = 20 \implies 4x + 10 = 20 \implies 4x = 10 \implies x = \frac{5}{2}$ Hence, the intersection is $\left(\frac{5}{2}, 2\right)$.

Step 2: Vertices of the Triangle

The vertices of the triangle are:

1. $(1, 2)$
2. $\left(1, \frac{16}{5}\right)$
3. $\left(\frac{5}{2}, 2\right)$

Step 3: Use the area formula for a triangle

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by: $\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substitute the coordinates:

• $(x_1, y_1) = (1, 2)$
• $(x_2, y_2) = \left(1, \frac{16}{5}\right)$
• $(x_3, y_3) = \left(\frac{5}{2}, 2\right)$

$\text{Area} = \frac{1}{2} \left| 1\left(\frac{16}{5} - 2\right) + 1\left(2 - 2\right) + \frac{5}{2}\left(2 - \frac{16}{5}\right) \right|$

Simplify each term:

1. $\frac{16}{5} - 2 = \frac{16}{5} - \frac{10}{5} = \frac{6}{5}$
2. $2 - 2 = 0$
3. $2 - \frac{16}{5} = \frac{10}{5} - \frac{16}{5} = -\frac{6}{5}$

Substitute: $\text{Area} = \frac{1}{2} \left| 1 \cdot \frac{6}{5} + 1 \cdot 0 + \frac{5}{2} \cdot \left(-\frac{6}{5}\right) \right|$

Simplify further: $\text{Area} = \frac{1}{2} \left| \frac{6}{5} - \frac{30}{10} \right| = \frac{1}{2} \left| \frac{6}{5} - \frac{15}{5} \right| = \frac{1}{2} \left| -\frac{9}{5} \right|$

$\text{Area} = \frac{1}{2} \cdot \frac{9}{5} = \frac{9}{10}$

Final Answer

The area of the triangle is: $\boxed{\frac{9}{10}}$

Let me know if you want further explanation or assistance with another question! Here are some related problems:

1. How to calculate the area of a triangle given the equation of its sides?
2. What is the formula for finding the intersection of two lines?
3. How do you determine the type of triangle formed by given lines?
4. What happens if the three given lines do not form a triangle?
5. How can you extend this method for triangles in 3D space?

Tip: Always check if the lines truly intersect to form a closed triangle before calculating the area.