To find the lengths of the sides of a triangle, we can use something called the Distance Formula. This formula helps us measure how far apart two points are on a graph.

Here’s the formula:

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

Easy Steps to Use the Distance Formula

1. Name Your Points: First, you need to label the corners (or vertices) of your triangle. Let's say we have triangle ABC with points A(1, 2), B(4, 6), and C(5, 1).

2. Find the Lengths of Each Side: Now, we will use the Distance Formula to calculate the lengths of each side.

   • For side AB:

     $$AB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$

   • For side BC:

     $$BC = \sqrt{(5 - 4)^2 + (1 - 6)^2} = \sqrt{1^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26}$$

   • For side CA:

     $$CA = \sqrt{(5 - 1)^2 + (1 - 2)^2} = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}$$

3. Understand the Triangle: After finding the lengths, we have AB = 5, BC is about 5.1, and CA is about 4.1. Now, you can see what type of triangle it is, like a scalene or isosceles, and you can check if it follows the triangle inequality (which means the sum of the lengths of any two sides should be greater than the length of the third side).

By practicing these steps, you’ll get better at using the Distance Formula and understanding triangles in geometry!