Pythagorean theorem and distance Knowledge-Based Study

Study notes

1. Fundamentals of the Theorem

   In any right-angled triangle, the side opposite the right angle is called the hypotenuse, and it is always the longest side. The two sides that form the right angle are called the legs.

   If the lengths of the legs are denoted by $a$ and $b$, and the length of the hypotenuse is denoted by $c$, the Pythagorean theorem states that:

   $$a^2 + b^2 = c^2$$

   This fundamental geometric relation allows us to find the length of any side of a right triangle when the lengths of the other two sides are known:

   • To find the hypotenuse: $c = \sqrt{a^2 + b^2}$
   • To find a missing leg: $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$

   Constructing Right Triangles on a Grid

   When two points, $A(x_1, y_1)$ and $B(x_2, y_2)$, lie on a Cartesian coordinate plane, the straight-line segment joining them represents the shortest path between them.

   We can construct a right-angled triangle by introducing a third point, $C(x_2, y_1)$, which shares the horizontal coordinate of point $B$ and the vertical coordinate of point $A$:

   • The horizontal leg $AC$ has length $|x_2 - x_1|$.
   • The vertical leg $BC$ has length $|y_2 - y_1|$.
   • The segment $AB$ serves as the hypotenuse of the triangle.

   Because the coordinate axes are perpendicular, triangle $ABC$ is guaranteed to be a right triangle with the right angle at vertex $C$.

   From Geometric Legs to Coordinate Distance

   Applying the Pythagorean theorem directly to the right triangle formed on the coordinate grid gives:

   $$(AB)^2 = (AC)^2 + (BC)^2$$

   Substituting the coordinate expressions for the lengths of the legs:

   $$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$

   Taking the positive square root of both sides yields the distance formula:

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

   Because squaring any real number produces a non-negative result, the order of subtraction does not affect the outcome: $(x_2 - x_1)^2 = (x_1 - x_2)^2$ and $(y_2 - y_1)^2 = (y_1 - y_2)^2$.

   Step-by-Step Calculation

   Calculate the distance between points $P(-2, 3)$ and $Q(4, -5)$.

   1. Identify coordinates: Let $x_1 = -2$, $y_1 = 3$, $x_2 = 4$, and $y_2 = -5$.
   2. Find horizontal change: $\Delta x = x_2 - x_1 = 4 - (-2) = 6$.
   3. Find vertical change: $\Delta y = y_2 - y_1 = -5 - 3 = -8$.
   4. Square both differences: $(\Delta x)^2 = 6^2 = 36$ and $(\Delta y)^2 = (-8)^2 = 64$.
   5. Sum the squares: $36 + 64 = 100$.
   6. Take the square root: $d = \sqrt{100} = 10$.

   Thus, the straight-line distance between $P$ and $Q$ is exactly 10 units.