Question

distance pythrgorean theroem

Answer 1

The Pythagorean theorem is a fundamental principle in geometry that relates to right-angled triangles. It states that, in a right-angled triangle, the square of the length of the hypotenuse \(c\) (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides \(a\) and \(b\). This can be expressed mathematically as:

\[
c^2 = a^2 + b^2
\]

### Calculating Distance Using the Pythagorean Theorem

The theorem can also be applied to find the distance between two points in a Cartesian coordinate system. If you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the distance \(d\) between these two points can be calculated using the formula derived from the Pythagorean theorem:

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

### Example

Suppose you want to find the distance between the points \((3, 4)\) and \((7, 1)\):

1. Identify the coordinates:
- \(x_1 = 3\), \(y_1 = 4\)
- \(x_2 = 7\), \(y_2 = 1\)

2. Apply the distance formula:
\[
d = \sqrt{(7 - 3)^2 + (1 - 4)^2}
\]
\[
d = \sqrt{(4)^2 + (-3)^2}
\]
\[
d = \sqrt{16 + 9}
\]
\[
d = \sqrt{25}
\]
\[
d = 5
\]

Thus, the distance between the points \((3, 4)\) and \((7, 1)\) is \(5\) units.