Question

How are the Pythagorean Theorem and the distance formula related?

Answer 1

Hint: We need to know the definition of the Pythagorean Theorem and the definition for the distance formula. Also, we need to know how to find the similarities between the Pythagorean Theorem and Distance formula. Also, this question involves the arithmetic operations like addition/ subtraction/ multiplication/ division.

Complete step by step solution:
The distance formula is shown below,
\[D = \sqrt {{{\left( {{x_1} - {x_2}} \right)}^2} + {{\left( {{y_1} - {y_2}} \right)}^2}} \]
The Pythagorean Theorem is shown below,
\[c = \sqrt {{a^2} + {b^2}} \]
Consider the following major points (in Euclidean geometry on a Cartesian coordinate axis),
The definition of a distance \[x\] \[ \pm c\] is \[\left| {x - c} \right|\]
There is the relationship where,
\[\sqrt {{{\left( {x - c} \right)}^2}} = \left| {x - c} \right| = x - c\]and\[ - x + c\]
The distance from one point to another point is the definition of a line segment.
Any diagonal line segment has a\[x\] component and a \[y\] component because a slope
is \[\dfrac{{\Delta y}}{{\Delta x}}\]. The greater the \[y\] contribution, the steeper the slope. The greater the \[x\] contribution, the flatter the slope.
We should see that these are two formulas relating to the diagonal distance definitions above. Let’s take
\[
{x_1} - {x_2} = \pm a \\
{y_1} - {y_2} = \pm b \\
\]
Now we can see the equivalence.
\[D = \sqrt {{{\left( { \pm a} \right)}^2} + {{\left( { \pm b} \right)}^2}} = c\]
\[D = \sqrt {{a^2} + {b^2}} \]
In short, the distance formula is a formalization of the Pythagorean Theorem
using \[x\] and \[y\] coordinates. So, they are the same thing in different contexts.

Note: Remember the definition and expression for the distance formula and the definition and expression for the Pythagoras theorem to make an easy calculation. Also, to solve these types of questions we would find the similarities between the given terms with the help of their definition and expressions.