Find the length of the diagonal of a triangle if its sides are 4 and 9.

Let a rectangle ABCD be given, in which AB = CD = 4, AD = BC = 9, AC = BD are diagonals.

The diagonals of a rectangle are equal to each other, and one diagonal divides the rectangle into two equal right-angled triangles.

In rectangle ABCD, the diagonal AC divides it into triangles △ ABC (∠B = 90˚) and △ CDA (∠D = 90˚).

Consider △ ABC: ∠B = 90˚, AB = 4 and BC = 9 – legs, AC – hypotenuse (since it lies opposite the right angle).

By the Pythagorean theorem:

AC = √ (AB² + BC²) = √ (4² + 9²) = √ (16 + 81) = √97.

Answer: AC = √97.