Line segments

In coordinate geometry, we often need to find the length of a line segment between two points and the area between curves and the x-axis. While calculus is often used to find areas under curves, we will focus on using simpler methods like the triangle area formula for areas between lines and the x-axis.

1. Finding the Length of a Line Segment:

To find the length of a line segment between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ on the coordinate plane, we use the distance formula. The formula is derived from the Pythagorean theorem and is given by:

$$\text{Length}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Example:

Given the points $A(2,3)$ and $B(6,7)$, we can find the length of the line segment between them:

$$\text{Length}=\sqrt{(6-2{)}^2+(7-3{)}^2}=\sqrt{4^2+4^2}=\sqrt{16+16}=\sqrt{32}=4\sqrt{2}\approx 5.66$$

2. The Area Between Equations and the X-Axis (Using the Triangle Formula):

To find the area between a line and the x-axis, we can use the formula for the area of a triangle. If we have a line that intersects the x-axis at some point, the area under the line forms a triangle. The area of a triangle is given by:

$$\text{Area}=\frac{1}{2}\times \text{Base}\times \text{Height}$$

Example 1:

Consider the line $y=2x$ between $x=0$ and $x=3$. To find the area between this line and the x-axis, we first determine the base and height of the triangle:

• The base of the triangle is the horizontal distance between $x=0$ and $x=3$, so the base is 3.
• The height is the value of $y$ when $x=3$, which is $y=2(3)=6$.

Now, apply the triangle area formula:

$$\text{Area}=\frac{1}{2}\times 3\times 6=\frac{1}{2}\times 18=9$$

So, the area under the line $y=2x$ between $x=0$ and $x=3$ is 9 square units.

Example 2:

Consider the line $y=-x+4$ between $x=0$ and $x=4$. We again apply the triangle formula:

• The base of the triangle is 4 (from $x=0$ to $x=4$).
• The height is the value of $y$ when $x=4$, which is $y=-(4)+4=0$.

So, we need to find the height when $x=0$, which is $y=4$. Now, applying the formula:

$$\text{Area}=\frac{1}{2}\times 4\times 4=\frac{1}{2}\times 16=8$$

Therefore, the area under the line $y=-x+4$ between $x=0$ and $x=4$ is 8 square units.

3. Summary:

• To find the length of a line segment between two points, use the distance formula: $$\text{Length}=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$
• To find the area between a line and the x-axis, use the triangle area formula: $$\text{Area}=\frac{1}{2}\times \text{Base}\times \text{Height}$$