Question

Draw the graphs of the equations x + 2y = 4 and 3x – 2y = 4. Find the area of triangle formed by the lines and x-axis.

Answer 1

1. Find the vertices of the triangle

The triangle is formed by the intersection of the two given lines and their respective intersections with the x-axis, where y = 0.

Intersection of the two lines:

To find where x + 2y = 4 and 3x – 2y = 4 meet, add the two equations together:

(x + 2y) + (3x – 2y) = 4 + 4

4x = 8

⇒ x = 2

Substitute x = 2 into x + 2y = 4:

2 + 2y = 4

⇒ 2y = 2

⇒ y = 1

The intersection point apex of the triangle is (2, 1).

Intersection with the x-axis (y = 0):

For x + 2y = 4:

If y = 0, then x = 4.

Point: (4, 0)

For 3x – 2y = 4:

If y = 0, then 3x = 4

⇒ `x = 4/3 ≈ 1.33`

Point: `(4/3, 0)`

2. Draw the graphs

To draw the lines, plot the intercepts and the intersection point found above:

Line 1 (x + 2y = 4):

Passes through (0, 2) and (4, 0).

Line 2 (3x – 2y = 4):

Passes through (0, –2) and (2, 1).

3. Calculate the area

The area of a triangle is given by the formula:

Area = `1/2` × base × height

Base: The distance between the two x-intercepts on the x-axis.

Base = `4 - 4/3`

= `(12 - 4)/3`

= `8/3 ≈ 2.67` units

Height: The y-coordinate of the intersection point (2, 1).

Height = 1 unit

Area calculation:

Area = `1/2 xx 8/3 xx 1`

= `4/3 ≈ 1.33` sq units

The area of the triangle formed by the lines and the x-axis is 1.33 square units or `4/3` sq units.