4.1. Formula

The distance 𝑑 between two points 𝐴(𝑥₁,𝑦₁) and 𝐵(𝑥₂,𝑦₂) is given by:

𝑑 = √((𝑥₂ − 𝑥₁)² + (𝑦₂ − 𝑦₁)²)

The formula to calculate the distance between two points in a Cartesian plane is derived from the Pythagorean theorem. In this formula, (𝑥₁, 𝑦₁) and (𝑥₂, 𝑦₂) represent the coordinates of the two points. To find the distance, you subtract the x-coordinates and the y-coordinates, square both results, add these squares together, and then take the square root of the sum. This gives you the straight-line distance between the two points.

Imagine you are standing on a grid map and you want to know how far away your friend is if they are also standing on the map. If you know their exact location (like their house's coordinates), you can use this formula to find out how many steps you would need to take in a straight line to get to them.

Find the distance between 𝐴(2,3) and 𝐵(6,7):

𝑑 = √((6−2)² + (7−3)²) = √(16 + 16) = √32 = 4√2

In this example, we are calculating the distance between two specific points. First, we substitute the coordinates into the distance formula: A is (2, 3) and B is (6, 7). We calculate the difference of the x-coordinates (6 - 2) and the y-coordinates (7 - 3) which gives us 4 and 4 respectively. Squaring these differences gives us 16 and 16. Adding these results together gives us 32. Finally, we take the square root of 32, leading us to the final answer of 4√2, which is the distance between points A and B.

Think of it like measuring the distance between two street intersections on a city map. By using the coordinates as the addresses of these intersections, you can easily calculate how far apart they are using the same method as when you measure distance using a ruler.

The midpoint 𝑀 of a line segment joining 𝐴(𝑥₁,𝑦₁) and 𝐵(𝑥₂,𝑦₂):

𝑀 = ( (𝑥₁ + 𝑥₂) / 2 , (𝑦₁ + 𝑦₂) / 2 )

The midpoint formula helps us find the exact middle point between two points on a line segment. By taking the average of the x-coordinates and the y-coordinates separately, we can pinpoint the location of the midpoint. This is useful in various geometrical constructions and real-world applications.

Imagine you are sharing a pizza with a friend. The point where you both meet to share the pizza is like the midpoint between two points. If you both decide to sit at equal distances from each other on the table, you can easily find that perfect midway point using the coordinates just like in this formula.

Midpoint of 𝐴(2,3) and 𝐵(6,7):

𝑀 = ( (2+6)/2 , (3+7)/2 ) = (4,5)

To find the midpoint M between points A(2,3) and B(6,7), we will add the x-coordinates together (2 + 6) and divide by 2, which gives us 4. For the y-coordinates, we do the same: (3 + 7) / 2 results in 5. Therefore, the midpoint M is at the coordinates (4, 5), indicating the halfway point on the line segment joining A and B.

If you have two stops on a road trip where one is at 2 miles, and the other is at 6 miles, the average distance you would travel to find the halfway point of your journey would be 4 miles. Just like with our midpoint formula, you find where you balance between both stops.

The gradient 𝑚 of the line through 𝐴(𝑥₁,𝑦₁) and 𝐵(𝑥₂,𝑦₂):

𝑚 = (𝑦₂ − 𝑦₁) / (𝑥₂ − 𝑥₁)

The gradient, often referred to as the slope, measures how steep a line is. It represents the change in the y-coordinate (rise) for a change in the x-coordinate (run). By using this formula, we can determine whether the line rises, falls, is horizontal, or vertical. A positive gradient means the line rises as it moves from left to right, while a negative gradient means it falls.

You can think of the gradient like a hill. If you're climbing up a steep hill, the gradient is positive - you're gaining altitude. If you’re going down a hill, the gradient is negative - you're losing altitude. A flat road has a gradient of zero.

Gradient between 𝐴(1,2) and 𝐵(4,6):

𝑚 = (6−2) / (4−1) = 4 / 3

In this example, we find the gradient m between two points A(1,2) and B(4,6) by subtracting the y-coordinate of point A from that of point B, resulting in 4. For the x-coordinates, we do 4 - 1, yielding 3. By dividing 4 by 3, we determine that the gradient of the line is 4/3, indicating it rises as we move from point A to point B.

Consider this like walking up a hill – if it takes you 4 steps up for every 3 steps you take horizontally, you're moving up at a 4/3 gradient. That means the hill is on a gentle slope instead of being too steep or flat.

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