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Distance Between Two Points




Understanding the Distance Between Two Points

Today, we’ll dive into a fundamental concept of geometry – the distance between two points. Whether on a flat plane or a map, knowing how to calculate this distance is essential in mathematics and various real-world applications.

What Is Distance?

In the simplest terms, distance is a measure of how far apart two points are from each other. Imagine you’re drawing a straight line from one point to another – this line represents the shortest path between them, and the length of this line is what we call the distance.

The Distance Formula

To calculate the distance between two points on a coordinate plane, we use the distance formula derived from the Pythagorean theorem. The Pythagorean theorem tells us that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

The distance formula is:

d = √((x2 - x1)² + (y2 - y1)²)

Where d represents the distance, (x1, y1) represents the coordinates of the first point, and (x2, y2) represents the coordinates of the second point.

Example Calculation

Let’s put this formula to work with an example. Suppose we have two points, A and B, with coordinates A(3, 4) and B(7, 1). To find the distance between them:

  1. First, identify the coordinates: x1 = 3, y1 = 4, x2 = 7, y2 = 1.
  2. Next, plug these coordinates into the formula:
d = √((7 - 3)² + (1 - 4)²)
d = √((4)² + (-3)²)
d = √(16 + 9)
d = √25
d = 5

So, the distance between points A and B is 5 units.

Potential Questions from Students

  1. Q: What if the points have negative coordinates?
  2. A: Negative coordinates work the same way as positive ones. Remember that when you square a negative number, it becomes positive.
  3. Q: Do we always have to draw a graph to find the distance?
  4. A: No, the graph is not necessary. It’s simply a visual aid. The formula works perfectly with just the coordinates.

Class Problems

Now, let’s practice with a few problems:

  1. Find the distance between the points C(-2, -3) and D(1, 4).
  2. Calculate the distance from E(5, -5) to F(-1, 3).
  3. Determine the distance between G(0, 0) and H(8, 6).

Try to solve these on your own before looking at the solutions below!

Solutions

  1. The distance between C and D: 
    d = √((-2 - 1)² + (-3 - 4)²)
d = √((-3)² + (-7)²)
d = √(9 + 49)
d = √58
d ≈ 7.62 (rounded to two decimal places)
  2. The distance from E to F: 
    d = √((5 - (-1))² + (-5 - 3)²)
d = √((6)² + (-8)²)
d = √(36 + 64)
d = √100
d = 10
  3. The distance between G and H: 
    d = √((0 - 8)² + (0 - 6)²)
d = √((-8)² + (-6)²)
d = √(64 + 36)
d = √100
d = 10

Keep practicing, and soon finding the distance between two points will be second nature to you.

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