Figs. 1.6–1.9

Given:

A$\left(3,4\right)$, D$\left(7,1\right)$, M$\left(9,6\right)$.

To Find:

AD and a general distance rule.

Property:

Baudhāyana–Pythagoras applies to the perpendicular legs.

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

Solution:

The signed changes A → D are Δx = 4 and Δy = −3. Leg lengths are 4 and 3.

${\mathrm{AD}}^2$ = $4^2$ + $3^2$ = 25, so AD = 5 units.

Similarly, DM = $\sqrt{29}$ and MA = $\sqrt{40}$.

In general, square both coordinate differences and add; take the non-negative square root.

Reflecting in the y-axis sends A to $\left(-3,4\right)$, D to $\left(-7,1\right)$, M to $\left(-9,6\right)$. Horizontal changes reverse sign; squared changes and lengths stay the same.

Reflection in the x-axis instead reverses y-coordinates, again preserving lengths.

Results:

AD = 5 units; reflections preserve all three side lengths.