How do you determine the distance between (0, 0) and (5, 12)?

You determine the distance using the Pythagorean theorem, which calculates the hypotenuse of a right-angled triangle.

To find the distance between the points (0, 0) and (5, 12), you can visualise these points on a coordinate grid. The point (0, 0) is the origin, and the point (5, 12) is 5 units to the right and 12 units up from the origin. These two points form a right-angled triangle with the x-axis and y-axis.

In this triangle, the horizontal leg (along the x-axis) is 5 units long, and the vertical leg (along the y-axis) is 12 units long. According to the Pythagorean theorem, the square of the hypotenuse (the distance between the two points) is equal to the sum of the squares of the other two sides.

So, you calculate the distance (d) as follows:
\[ d = \sqrt{(5^2 + 12^2)} \]

First, square the lengths of the legs:
\[ 5^2 = 25 \]
\[ 12^2 = 144 \]

Next, add these squares together:
\[ 25 + 144 = 169 \]

Finally, take the square root of the sum to find the hypotenuse:
\[ d = \sqrt{169} = 13 \]

Therefore, the distance between the points (0, 0) and (5, 12) is 13 units. This method can be used for any two points on a coordinate plane by substituting the appropriate coordinates into the Pythagorean theorem.