Answer:
|AB|² − |CD|² = |BC|² − |AD|²
Step-by-step explanation:
The quadrilateral has two right angles (A and C) opposite of each other, so the diagonal splits the quadrilateral into two right triangles. Since these are right triangles, we can use Pythagorean theorem (a² + b² = c²), which says the sum of the squares of the perpendicular sides (a and b) is equal to the square of the remaining side (c), called the hypotenuse.
For the first triangle:
|AB|² + |AD|² = |BD|²
For the second triangle:
|BC|² + |CD|² = |BD|²
Setting them equal:
|AB|² + |AD|² = |BC|² + |CD|²
Subtract |AD|² and |CD|² from both sides:
|AB|² − |CD|² = |BC|² − |AD|²
